(B) According to Section 7.2 or [YaZh2021, Section 2], we can make a conformal change of coordinates so that, in a neighborhood of y, the boundary $\partial\Omega$ is a segment centered at y. A neighborhood of y then contains a set of the form $D^+(r_2) := D(y, r_2) \cap \Omega$ (for these local coordinates), and we also take some $r_0 < r_1 < r_2$ to be chosen later on.

Taking $r_1 < r_2$ small enough, we may assume that the following properties hold.

(B-a) The set $D^+(r_1)$ only contains one singular point of $w$, namely the point $y$, i.e., $D^+(r_1) \cap S_i(w) = \emptyset$ and $D^+(r_1) \cap S_b(w) = \{y\}$, so that the function $\tilde{w}$ defined in (7.13) only vanishes at $y$ on the set $\bar{D}^+(r_1) \cap \Gamma$.

(B-b) Let $\mu_1(D^+(r_1))$ denote the least eigenvalue of $-\Delta + V$ in the domain $D^+(r_1)$ with mixed boundary condition, Dirichlet on $\partial D^+(r_1) \cap \Omega$, and the current boundary condition (3.4) on $\partial D^+(r_1) \cap \Gamma$ (Dirichlet or Robin). Choose $r_1$ small enough so that $\mu_1(D^+(r_1)) > \lambda_k$ ($\lambda_k$ is the eigenvalue associated with $U$). This is possible because $\mu_1(D^+(r))$ tends to infinity when $r$ tends to 0 (use an extended monotonicity property for $\mu_1$ in such domains$^6$, and comparison with the eigenvalue of a half disk with Neumann condition on the diameter, and Dirichlet condition on the half circle).

(B-c) In the set $D^+(r_1)$, the local structure theorem applies to the function $w$. More precisely, let $(r, \omega)$ be local polar coordinates centered at $y$, with respect to a direct frame $\{\vec{e}_1, \vec{e}_2\}$ such that $\vec{e}_1$ is tangent to $\Gamma$ at $y$, and $\vec{e}_2$ is normal to $\Gamma$, pointing inwards.

$\diamond$ In the Dirichlet case, $w(r, \omega) = c_w r^p \sin(p\omega) + \mathcal{O}(r^{p+1})$. Define the rays $\{\omega = \omega_j \mid 1 \le j \le p-1\}$, where $\omega_j := j\frac{\pi}{p}$. Then, in $D^+(r_1)$, the nodal set $\mathcal{Z}(w)$ consists of $(p-1)$ arcs $r \mapsto \delta_j(r) := (r, \tilde{\omega}_j(r))$, where the functions $\tilde{\omega}_j(r)$ are smooth for $0 < r < r_1$, with $\lim_{r\to 0} \tilde{\omega}_j(r) = \omega_j$. These arcs are transverse to the circles $\{r=c\}$ with $c < r_1$. Note that $\rho(w, y) = q = (p-1)$.

$\diamond$ In the Robin case, $w(r, \omega) = c_w r^p \cos(p\omega) + \mathcal{O}(r^{p+1})$. Define the rays

$$ \{\omega = \omega_j \mid 1 \le j \le p\}, $$

where $\omega_j := (j - \frac{1}{2})\frac{\pi}{p}$. Then, in $D^+(r_1)$, the nodal set $\mathcal{Z}(w)$ consists of $p$ arcs $r \mapsto \delta_j(r) := (r, \tilde{\omega}_j(r))$, where the functions $\tilde{\omega}_j(r)$ are smooth for $0 < r < r_1$, with $\lim_{r\to 0} \tilde{\omega}_j(r) = \omega_j$. These arcs are transverse to the circles $\{r=c\}$ with $c < r_1$. Note that $\rho(w, y) = q = p$.

(B-d) Let $\delta(\omega) := (r_0, \omega)$ be a parametrization of $C^+(r_0) := \partial D(r_0) \cap \Omega$. Then, $C^+(r_0) \cap \mathcal{Z}(w) = \{A_1(r_0), \dots, A_q(r_0)\}$. Choose $0 < \varepsilon_1 < 1$, small enough so that, for $1 \le j \le q$, the open arcs $(A_j(r_0) - \varepsilon_1, A_j(r_0) + \varepsilon_1)$ of $C^+(r_0)$, with length $2\varepsilon_1$, centered at $A_j(r_0)$ are pairwise disjoint. Define the set $C^+(r_0, w, \varepsilon_1)$ by

$$ C^+(r_0, w, \varepsilon_1) := \bigcup_{j=1}^{q} (A_j(r_0) - \varepsilon_1, A_j(r_0) + \varepsilon_1), $$

<footnote>$^6$The monotonicity follows from the min-max principle and the fact that the piece of boundary possibly with the Robin condition is always contained in $\Gamma$.</footnote>
see Figure 9.1. Call $A_0(r_0)$ and $A_{q+1}(r_0)$ the extremities of $C^+(r_0) \cap \Gamma$. Since the (sub)sequence $\mathcal{Z}(u_n)$ tends to $\mathcal{Z}(w)$ in the Hausdorff distance, for $n$ large enough, we have $\mathcal{Z}(u_n) \cap C^+(r_0) \subset C^+(r_0, w, \varepsilon_1)$.

<figure><img src="image_27.png" /><figcaption>FIGURE 9.1. Proof of Lemma 9.2: the set $C^+(r_0, w, \varepsilon_1)$</figcaption></figure>

(C) From now on, we assume that $\{x_n\} \subset D^+(r_0/2)$, and that $n$ is large enough so that the above Property (B-d) is satisfied. Taking Lemma 8.1 into account, there are two possible cases.

**Case C1.** There exists an infinite subsequence $\{u_{s(n)}\}$ of the sequence $\{u_n\}$ such that, for all $n$, $S_b(u_{s(n)}) = \emptyset$.

In this case, according to the proof of Lemma 8.1, the nodal set $\mathcal{Z}(u_{s(n)})$ consists of $(k-1)$ simple loops at $x_{s(n)}$; these loops do not intersect away from $x_{s(n)}$, and do not hit $\Gamma$. Choose $\gamma$, any of these loops. Since $x_{s(n)} \in D^+(r_0/2)$, either the loop $\gamma$ crosses $C^+(r_0)$ at (at least) two distinct points $z_{\gamma,1}$ and $z_{\gamma,2}$, or its is entirely contained in $\bar{D}^+(r_0)$. In the latter case, the function $u_{s(n)}$ would have a nodal domain contained in $D^+(r_0)$, and this would contradict the above Property (B-b) by the monotonicity principle for Dirichlet eigenvalues, see Figures 9.2 (right) and 9.3 (forbidden configurations). For each $n$, the set $\mathcal{Z}(u_{s(n)})$ consists of $(k-1)$ loops which do not intersect away from $x_{s(n)}$. It follows that we have at least $2(k-1)$ distinct points $z_{s(n),j}$ in $C^+(r_0) \cap \mathcal{Z}(u_{s(n)})$, for $1 \le j \le (2k-2)$. Since $\{\mathcal{Z}(u_{s(n)})\}$ converges to $\mathcal{Z}(w)$ in the Hausdorff distance, the sequences $\{z_{s(n),j}\}$, $1 \le j \le (2k-2)$, converge to points $\zeta_j \in C^+(r_0, w, \varepsilon_1) \cap \mathcal{Z}(w) = \{A_j(r_0), 1 \le j \le q\}$.

<figure><img src="image_28.png" /><figcaption>FIGURE 9.2. Proof of Lemma 9.2: Case C1, nodal pattern.</figcaption></figure>
<figure><img src="image_29.png" /><figcaption>FIGURE 9.3. Proof of Lemma 9.2: Case C1, forbidden configurations.</figcaption></figure>

**Claim 9.4.** The points $\zeta_j, 1 \le j \le (2k-2)$, are pairwise distinct.

*Proof.* Indeed, assume this is not the case, say $\zeta_1 = \zeta_2$. Then, the two sequences $\{z_{s(n),1}\}$ and $\{z_{s(n),2}\}$, with $z_{s(n),1} \neq z_{s(n),1}$, would converge to the same $A_j(r_0)$ for some $j, 1 \le j \le q$. Writing $z_{s(n),i} = \delta(\theta_{s(n),i})$, we would find some $\theta_{s(n),3}$ in the smaller arc between $\theta_{s(n),1}$ and $\theta_{s(n),2}$ such that $(u_{s(n)} \circ \delta)'(\theta_{s(n),3}) = 0$. Since $u_n$ tends to $u$ uniformly in $C^1$, and since $z_{s(n),i}$ tends to $A_j(r_0)$, $A_j(r_0)$ would be a singular point of the function $w$. Since the nodal rays of $w$ are transverse to $C^+(r_0)$, this would contradict the above Property (B-a). ✓

Claim 9.4 implies that $q \ge (2k-2)$, i.e., $\rho(w,y) \ge (2k-2)$. Lemma 8.3 implies that $w \in W_y$, and $\rho(w,y) = (2k-2)$, so that $y \in \Gamma_{(2k-2)}$.

**Remarks 9.5.**

(i) Note that Claim 9.4 also implies that, for $n$ large enough, each loop in $\mathcal{Z}(u_{s(n)})$ meets $C^+(r_0)$ at exactly two points, see Figure 9.2, right.

(ii) Note that Case C1 can only occur if $y \in \Gamma_{(2k-2)}$.

**Case C2.** There exists an infinite subsequence $\{u_{s(n)}\}$ such that, for each $n$, we have $S_b(u_{s(n)}) \neq \emptyset$.

In this case, according to the proof of Lemma 8.3, the nodal set $\mathcal{Z}(u_{s(n)})$ consists of $(k-2)$ simple loops at $x_{s(n)}$, and two simple nodal arcs from $x_{s(n)}$ to the boundary. The loops do not intersect away from $x_{s(n)}$, and do not hit $\Gamma$; the arcs do not intersect each other (except at $x_{s(n)}$, and possibly on $\Gamma$ if they hit $\Gamma$ at the same point) and do not intersect the loops. The energy argument used in Case C1, shows that each loop intersects $C^+(r_0)$ at (at least) two distinct points. A similar energy argument for mixed boundary conditions (Dirichlet on the nodal arcs, and the ambient boundary condition, Dirichlet or Robin, on $\Gamma$) shows that the nodal arcs cannot both be contained in $D^+(r_0)$, and at least one of them crosses $C^+(r_0)$, see Figure 9.4, and Figures 9.5 and 9.6 (forbidden configurations). Finally, for each $n$, we have at least $(2k-3)$ distinct points $z_{s(n),j}$ in $C^+(r_0) \cap \mathcal{Z}(u_{s(n)})$, for $1 \le j \le (2k-3)$. As in Case C1, these sequences converge to points $\zeta_j \in \mathcal{Z}(w) \cap C^+(r_0)$. Applying Claim 9.4, we conclude that $q \ge (2k-3)$, i.e. that $\rho(w,y) \ge (2k-3)$ so that $w \in W_y$, and we have two possible cases, either $\rho(w,y) = (2k-3)$ and $y \in \Gamma_{(2k-3)}$, or $\rho(w,y) = (2k-2)$ and $y \in \Gamma_{(2k-2)}$.

At this stage we have proved that the only possible limit points of a sequence $\{u_n\}$ are in $W_y$, see Figure 9.2 (left). Since $\dim W_y = 1$, this proves Assertion (i) of the
lemma. Claim 9.4 also implies that, for $n$ large enough, $Z(u_{s(n)})$ meets $C^+(r_0)$ at precisely $(2k-3)$ or $(2k-2)$ points.

According to Lemma 8.3, we have $S_b(u_{s(n)}) = \{y_{s(n),1}, y_{s(n),2}\}$ possibly with $y_{s(n),1} = y_{s(n),2}$. The only possible limit points of these sequences are

$$ \begin{cases} y & \text{if } \rho(w, y) = (2k-2), \\ y \text{ and } z(y) & \text{if } \rho(w, y) = (2k-3). \end{cases} $$

When $\rho(w, y) = (2k-3)$, $\rho(w, z(y)) = 1$, and the function $\tilde{w}$ vanishes and changes sign at $z(y)$. Since $\{u_{s(n)}\}$ converges to $w C^1$-uniformly, this implies that, for $n$ large enough, the function $\tilde{u}_{s(n)}$ changes sign near $z(y)$, and hence that one sequence, say $\{y_{s(n),2}\}$ tends to $z(y)$, and the other $\{y_{s(n),1}\}$ tends to $y$. Note that they cannot both tend to $z(y)$ since $\rho(w, z(y)) = 1$.

When $\rho(w, y) = (2k-2)$, the sequences $\{y_{s(n),1}\}$ and $\{y_{s(n),2}\}$ must both converge to $y$.

Applying Claim 9.4, we find that there are three sub-cases.

C2(i): There exists a subsequence $\{u_{s(n)}\}$ such that $y_{s(n),1} = y_{s(n),2}$ tending to $y$. For energy reasons, the arcs from $x_{s(n)}$ to $y_{s(n),1}$ cannot both be contained in $D^+(r_0)$. One of these arcs, and actually only one for $n$ large enough, has to meet $C^+(r_0)$ at two distinct points, see Remark 9.5(i) and Figure 9.4 (left).

C2(ii): There exists a subsequence $\{u_{s(n)}\}$ such that $y_{s(n),1} \neq y_{s(n),2}$ both tending to $y$. For energy reasons, the arcs from $x_{s(n)}$ to $y_{s(n),1}$ and $y_{s(n),2}$ cannot both be contained in $D^+(r_0)$. One of these arcs, and actually only one for $n$ large enough, has to meet $C^+(r_0)$ at two distinct points. See Figure 9.4 (center).

C2(iii): There exists a subsequence $\{u_{s(n)}\}$ such that $y_{s(n),1} \neq y_{s(n),2}$, with $y_{s(n),1}$ tending to $y$ and $y_{s(n),2}$ tending to some $z \neq y$. For $n$ large enough, the arc from $x_{s(n)}$ to $y_{s(n),2}$ intersects $C^+(r_0)$ at one point, and the arc from $x_{s(n)}$ to $y_{s(n),1}$ stays inside $D^+(r_0)$. See Figure 9.4 (right).

In subcases C2(i) and C2(ii), we have $\rho(w, y) = (2k-2)$, so that $y \in \Gamma_{(2k-2)}$. In subcase C2(iii), we have $\rho(w, y) = (2k-3)$, so that $y \in \Gamma_{(2k-3)}$ with $z(y) = z$, the limit of $\{y_{s(n),2}\}$.

<figure><img src="image_30.png" /><figcaption>FIGURE 9.4. Proof of Lemma 9.2: Case C2, nodal patterns</figcaption></figure>

This proves Assertions (ii) and (iii). The proof of Lemma 9.2 is complete. $\square$
<figure><img src="image_31.png" /><figcaption>FIGURE 9.5. Proof of Lemma 9.2: Case C2i, forbidden configurations</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>FIGURE 9.6. Proof of Lemma 9.2: Case C2, forbidden configurations</figcaption></figure>

Figures 9.5 and 9.6 display forbidden configurations for $\mathcal{Z}(u_n)$ when $r_1$ is small enough and $n$ large: a loop cannot be contained in $D^+(r_0)$; the arcs cannot both be contained in $D^+(r_0)$; the arcs cannot both meet $C^+(r_0)$.

**Remark 9.6.** The nodal patterns displayed in the above figures hold for both the Dirichlet and Robin boundary conditions. Unless otherwise stated this remark applies to all figures of this section.

Figures 9.2 and 9.4 display typical configurations for $\mathcal{Z}(u_n)$ when $r_1$ is small enough and $n$ large: the loops intersect $C^+(r_0)$ at two distinct points; one arc exits $D^+(r_0)$.

As a byproduct of the proof of Lemma 9.2, we obtain the configurations of the nodal sets $\mathcal{Z}(u_x)$ when $x \in \Omega$ tends to some $y \in \Gamma$. When $y \in \Gamma_{(2k-3)}$, we are in Case C2(iii). When $y \in \Gamma_{(2k-2)}$, one of the cases C1, C2(i) or C2(ii) occurs.

**9.2. Eigenfunctions with two prescribed boundary singular points.** Given $(y, s) \in \Gamma_{(2k-3)} \times \Gamma$, with $y \neq s$, define the subspace

$$ (9.2) \qquad V_{y,s} := \{u \in U \mid \rho(u, y) \ge 2k-4 \text{ and } \rho(u, s) \ge 1\}. $$

In view of the assumption (9.1), Lemma 3.11 implies that $V_{y,s} \neq \{0\}$.

In this subsection, we revisit Lemmas 3.4, 3.5 and 3.6 of [HoMN1999, pp. 1180-1183]. We retain the notation of the previous subsections and introduce the following one.

**Notation 9.7.** Given two points $y_1 \neq y_2 \in \Gamma$, we denote by $\mathcal{A}(y_1, y_2)$ the open arc from $y_1$ to $y_2$, moving counter-clockwise. Given $y \in \Gamma$ and $a$ smaller than half the length of $\Gamma$, we denote by $\mathcal{A}_c(y; a)$ the arc centered at $y$, with length $2a$, taken
counter-clockwise. In both cases, we use the mathematical symbols [ and ] to denote
the closed or semi-closed arcs.

9.2.1. *First properties of* $V_{y,s}$.

**Lemma 9.8.** Let $U := U(\lambda_k)$ with $k \ge 3$. Assume that $\Omega$ is simply connected, and $\dim U = (2k-2)$. Given $(y, s) \in \Gamma_{(2k-3)} \times \Gamma$, with $y \ne s$, the following properties of the subspace $V_{y,s}$ defined in (9.2) hold.

(i) The subspace $V_{y,s}$ has dimension 1. Let $v_{y,s}$ denote a generator of $V_{y,s}$.

(ii) Any $0 \neq u \in V_{y,s}$ satisfies

$$
(9.3) \qquad \left\{
\begin{array}{@{}l@{}}
\kappa(u) = k, \\
\mathcal{Z}(u) \cup \Gamma \text{ is connected,} \\
S_i(u) = \emptyset, \\
\sum_{z \in S_b(u)} \rho(u, z) = 2k-2, \text{ and} \\
\qquad 2k-4 \le \rho(u, y) \le 2k-3, \\
\qquad 1 \le \rho(u, s) \le 2.
\end{array}
\right.
$$

More precisely, there are three distinct possibilities.

**Case (1):** $\rho(u, y) = (2k-3)$ and $\rho(u, s) = 1$.

Case (2): $\rho(u, y) = (2k - 4)$ and $\rho(u, s) = 2$.

Case (3): $\rho(u, y) = (2k - 4)$, $\rho(u, s) = 1$, and there exists $s' \in \Gamma \setminus \{y, s\}$ such that $S_b(u) = \{y, s, s'\}$, $\rho(u, s') = 1$.

In Case (1), $u \in W_y$, and $s = z(y)$ (with the notation of Lemma 8.3).

(iii) If $s = z(y)$, then $V_{y,z(y)} = W_y$.

(iv) The map $\{(y,s) \mid (y,s) \in \Gamma_1 \times \Gamma, s \neq y\} \ni (y,s) \mapsto [V_{y,s}] \in \mathbb{P}(U)$ is $C^\infty$.

Proof. We already know that $\dim V_{y,s} \ge 1$.

We retain the notation of Lemma 8.3. In particular, for $y \in \Gamma_{(2k-3)}$, we have $W_y = [w_y]$ with $0 \ne w_y \in U$ satisfying $S_b(w_y) = \{y, z(y)\}$ with $z(y) \ne y$, and $\rho(w_y, y) = (2k-3)$, $\rho(w_y, z(y)) = 1$.

◇ *Proof of Assertion (ii).* From Euler's formula (8.2) we obtain,

$$
(9.4) \qquad 0 \ge \kappa(u) - k = (b_0(\mathcal{Z}(u) \cup \Gamma) - 1) + \frac{1}{2} \sum_{z \in S_i(u)} (\nu(u, z) - 2) \\
\qquad\qquad\qquad\qquad\qquad\qquad + \frac{1}{2} \left( \sum_{z \in S_b(u)} \rho(u, z) - 2k + 2 \right).
$$

If $0 \neq u \in V_{y,s}$, we have $\sum_{z \in S_b(u)} \rho(u, z) \geq 2k-3$, and hence $\sum_{z \in S_b(u)} \rho(u, z) \geq 2k-2$
since the sum is an even integer, by Corollary 3.8. All the terms in the right hand
side of (9.4) must vanish; this proves (9.3). Assertion (ii) then follows from (9.3)
and the assumptions that $\rho(u, y) \geq (2k-4)$ and $\rho(u, s) \geq 1$.

◇ *Proof of Assertion (i).*

(A) We first assume that $s \neq z(y)$. Assume that there are at least two linearly independent functions $u_1, u_2 \in V_{y,s}$, then $\rho(u_i, y) = (2k - 4)$ and $\rho(u_i, s) \ge 1$. According to Lemma 3.12, there exists a nontrivial linear combination $u$ of $u_1$ and $u_2$ such that $\rho(u, y) \ge 2k - 3$ and $\rho(u, s) \ge 1$. Euler's formula implies that $u$ pertains to Assertion (ii), Case (1), contradicting the fact that $s \neq z(y)$.
(B) We now assume that $s = z(y)$. In this case, a generator $w_y$ of $W_y$ belongs to $V_{y,z(y)}$. Assume that $\dim V_{y,z(y)} \ge 2$. Define $V'_{y,z(y)} = V_{y,z(y)} \ominus W_y$, which has dimension at least 1. If $\dim V_{y,z(y)} \ge 3$, we can find two linearly independent $u_1, u_2 \in V'_{y,z(y)}$, such that $\rho(u_i, y) = 2k-4$, and $\rho(u_i, s) \ge 1$. By Lemma 3.12, there exists a nontrivial linear combination $u \in V'_{y,z(y)}$ such that $\rho(u, y) \ge 2k-3$ and $\rho(u, s) \ge 1$. Hence, $u \in W_y$, a contradiction. Assuming that $\dim V_{y,z(y)} = 2$, we can choose a basis $\{w_y, v_y\}$ such that $v_y \notin W_y$. Then, $\rho(v_y, y) = 2k-4$, and there are two cases,

Case (a): $\rho(v_y, z(y)) = 2$,

Case (b): $\rho(v_y, z(y)) = 1$, and there exists some $z_1(y) \in \Gamma$, $z_1(y) \ne z(y)$, such that $S_b(v_y) = \{y, z(y), z_1(y)\}$ and $\rho(v_y, z_1(y)) = 1$.

Without loss of generality, making use of Lemma 3.14, we may choose the functions $w_y$ and $v_y$ as follows (we consider open arcs). First we choose $w_y$ so that $\tilde{w}_y > 0$ on the arc $\mathcal{A}(y, z(y))$, and $\tilde{w}_y < 0$ on the arc $\mathcal{A}(z(y), y)$.

◦ In Case (a), we choose $v_y$ such that $\tilde{v}_y > 0$ on $\mathcal{A}(y, z(y)) \cup \mathcal{A}(z(y), y)$.

◦ In Case (b), assuming that $z_1(y) \in \mathcal{A}(y, z(y))$, we choose $v_y$ such $\tilde{v}_y > 0$ on $\mathcal{A}(z(y), y) \cup \mathcal{A}(y, z_1(y))$, and $\tilde{v}_y < 0$ on $\mathcal{A}(z_1(y), z(y))$.

Figure 9.7 displays the signs of $\tilde{v}_y$ in both cases.

<figure><img src="image_33.png" /><figcaption>FIGURE 9.7. Signs of $\tilde{v}_y$, Cases (a) and (b)</figcaption></figure>

*Claim 1.* Under the assumption that $\dim V_{y,z(y)} = 2$, there exists some function $v \in V_{y,z(y)}$ such that $\rho(v, y) = 2k-4$ and $\rho(v, z(y)) = 2$.

*Proof of Claim 1.* If $v_y$ satisfies Claim 1, there is nothing to prove. If not, $v_y$ falls into Case (b) above.

Given $t \in \Gamma \setminus \{y, z(y)\}$, $\tilde{w}_y(t) \ne 0$, and we can define the function

$$ (9.5) \qquad \xi_t := a(t) w_y - b(t) v_y \in V_{y,z(y)}, $$

where

$$ (9.6) \qquad \left\{ \begin{array}{l} a(t) = \tilde{v}_y(t) (\tilde{v}_y^2(t) + \tilde{w}_y^2(t))^{-\frac{1}{2}}, \\ b(t) = \tilde{w}_y(t) (\tilde{v}_y^2(t) + \tilde{w}_y^2(t))^{-\frac{1}{2}}. \end{array} \right. $$

For $t \notin \{y, z(y)\}$, $b(t) \ne 0$, and hence $\rho(\xi_t, y) = 2k-4$, $\rho(\xi_t, z(y)) \ge 1$, $\rho(\xi_t, t) \ge 1$. Euler's formula applied to $\xi_t$ implies that $\rho(\xi_t, z(y)) = \rho(\xi_t, t) = 1$, and $S_b(\xi_t) = \{y, z(y), t\}$. According to Lemma 3.14, the function $\tilde{\xi}_t$ has precisely three zeros
at $y, z(y)$ and $t$, changes sign at $z(y)$ and $t$, and does not change sign at $y$. For $t \in A(z_1(y), z(y))$, $\xi_t(z_1(y)) < 0$, and we conclude that

$$ (9.7) \quad \text{for } t \in A(z_1(y), z(y)), \quad \begin{cases} \xi_t > 0 & \text{in } A(t, z(y)), \text{ and} \\ \xi_t < 0 & \text{in } A(z(y), y) \cup A(y, t). \end{cases} $$

where $y, z(y)$ and $z_1(y)$ are as in Figure 9.7(b).

Choose a sequence $\{t_n\} \subset A(z_1(y), z(y))$, with $t_n \to z(y)$. Taking a subsequence if necessary, we may assume that the sequence $\{(a(t_n), b(t_n))\}$ converges to some $(a, b) \in \mathbb{S}^1$, so that the sequence $\{\xi_{t_n}\}$ converges to the function $\xi := a w_y - b v_y$. From (9.7), we conclude that $\xi \le 0$ on $\Gamma$. Since $\xi \in V_{y,z(y)}$ we have three possibilities,

(i) $\rho(\xi, y) = 2k - 3$ and $\rho(\xi, z(y)) = 1$,

(ii) $\rho(\xi, y) = 2k - 4$, $\rho(\xi, z(y)) = 1$, and $\rho(\xi, z_2)$ for some $z_2 \ne y, z(y)$,

(iii) $\rho(\xi, y) = 2k - 4$ and $\rho(\xi, z(y)) = 2$.

Since (i) and (ii) are incompatible with $\xi \le 0$ on $\Gamma$, we conclude that $\rho(\xi, y) = 2k - 4$ and $\rho(\xi, z(y)) = 2$. This proves Claim 1. ✓

We now continue with part (B) in the proof of Assertion (i). In view of Claim 1, assuming that $\dim V_{y,z(y)} = 2$, we may choose a basis $\{w_y, v_y\}$ of $V_{y,z(y)}$ such that

$$ (9.8) \quad \left\{ \begin{array}{ll} \rho(w_y, y) = 2k-3, & \\ \rho(w_y, z(y)) = 1, & \text{and} \\ \check{w}_y|_{A(y,z(y))} > 0 \text{ and } \check{w}_y|_{A(z(y),y)} < 0, & \end{array} \right. \quad \left\{ \begin{array}{ll} \rho(v_y, y) = 2k-4, & \\ \rho(v_y, z(y)) = 2, & \\ \check{v}_y|_{\Gamma\{y,z(y)\}} > 0. & \end{array} \right. $$

Examples of nodal sets of these functions are displayed in Figure 9.8: on the left $\mathcal{Z}(w_y)$, on the right $\mathcal{Z}(v_y)$, with two possible cases.

<figure><img src="image_34.png" /><figcaption>FIGURE 9.8. Nodal sets of $w_y$ (left) and $v_y$ (right), with $k = 4$.</figcaption></figure>

From now on, to simplify the notation in the proof, we denote the arc $A(y, z(y))$ by $A_1$, and the arc $A(z(y), y)$ by $A_2$.

For $s \notin \{y, z(y)\}$, we consider the function

$$ (9.9) \qquad \xi_s = a(s) w_y - b(s) v_y, $$

where $w_y$ and $v_y$ satisfy (9.8), and $a(s)$, $b(s)$ are given by

$$ (9.10) \quad \left\{ \begin{array}{l} a(s) = \check{v}_y(s) (\check{v}_y^2(s) + \check{w}_y^2(s))^{-\frac{1}{2}}, \\ b(s) = \check{w}_y(s) (\check{v}_y^2(s) + \check{w}_y^2(s))^{-\frac{1}{2}}. \end{array} \right. $$

In particular, $a(s) > 0$ in $A_1 \cup A_2$, $b(s) > 0$ in $A_1$ and $b(s) < 0$ on $A_2$. Since
$a(s)$ and $b(s)$ are different from 0, $\rho(\xi_s, y) = (2k - 4)$ and $\rho(\xi_s, z(y)) = 1$. Since $\tilde{\xi}_s(s) = 0$, $\rho(\xi_s, s) \ge 1$. Since $\xi_s \in V_{y,z(y)}$, Equation (9.3) implies that $\rho(\xi_s, s) = 1$, $S_b(\xi_s) = \{y, z(y), s\}$, and $\tilde{\xi}_s$ changes sign at $z(y)$ and $s$ (use Lemma 3.14 again).

Taking $s_2 \in A_2$ and $s \in A_1$, we find that $\tilde{\xi}_s(s_2) < 0$ and hence,

$$ (9.11) \qquad \text{for } s \in A_1, \quad \begin{cases} \tilde{\xi}_s > 0 & \text{in } \mathcal{A}(s, z(y)), \\ \tilde{\xi}_s < 0 & \text{in } A_2 \cup \mathcal{A}(y, s). \end{cases} $$

Similarly, taking $s_1 \in A_1$ and $s \in A_2$, we find that $\tilde{\xi}_s(s_1) > 0$ and hence,

$$ (9.12) \qquad \text{for } s \in A_2, \quad \begin{cases} \tilde{\xi}_s < 0 & \text{in } \mathcal{A}(z(y), s), \\ \tilde{\xi}_s > 0 & \text{in } \mathcal{A}(s, y) \cup A_1. \end{cases} $$

<figure><img src="image_35.png" /><figcaption>FIGURE 9.9. Signs of the function $\tilde{\xi}_s$</figcaption></figure>

There exists a unique $\theta(s) \in (-\frac{\pi}{2}, \frac{\pi}{2})$ such that $a(s) = \cos(\theta(s))$ and $b(s) = \sin(\theta(s))$, so that $\xi_s = \cos(\theta(s))w_y - \sin(\theta(s))v_y$. Then, $\theta(s) \in (0, \frac{\pi}{2})$ when $b(s) > 0$ or equivalently when $s \in A_1$; $\theta(s) \in (-\frac{\pi}{2}, 0)$ when $b(s) < 0$ or equivalently when $s \in A_2$. Furthermore, the map $s \mapsto \theta(s)$ is injective because $S_b(\xi_s) = \{y, z(y), s\}$.

For $s_1, s_2 \in A_1$ or $A_2$, we have

$$ (9.13) \quad \left\{ \begin{aligned} \xi_{s_1} - \xi_{s_2} &= (\cos(\theta(s_1)) - \cos(\theta(s_2)))w_y - (\sin(\theta(s_1)) - \sin(\theta(s_2)))v_y \\ &= -2 \sin \frac{\theta(s_1) - \theta(s_2)}{2} \left( \sin \frac{\theta(s_1) + \theta(s_2)}{2} w_y + \cos \frac{\theta(s_1) + \theta(s_2)}{2} v_y \right). \end{aligned} \right. $$

If $s_1, s_2 \in A_1$, $\theta(s_1), \theta(s_2) \in (0, \frac{\pi}{2})$, and the second factor in the second line of Equation (9.13) is positive in $A_1$. If $s_1, s_2 \in A_2$, $\theta(s_1), \theta(s_2) \in (-\frac{\pi}{2}, 0)$, and the second factor in the second line of Equation (9.13) is positive in $A_2$.

Since $\tilde{\xi}_{s_1}(s_2) - \tilde{\xi}_{s_2}(s_2) = \tilde{\xi}_{s_1}(s_2)$, using Equations (9.11) and (9.12) we conclude that,

$$ (9.14) \quad \left\{ \begin{array}{l} s_1 \in A_1 \text{ and } s_2 \in \mathcal{A}(s_1, z(y)) \Rightarrow \tilde{\xi}_{s_1}(s_2) > 0 \\ \phantom{s_1 \in A_1 \text{ and }} \Rightarrow \sin \frac{\theta(s_1) - \theta(s_2)}{2} < 0 \quad \Rightarrow \theta(s_2) > \theta(s_1) \text{ in } (0, \frac{\pi}{2}), \\ s_1 \in A_2 \text{ and } s_2 \in \mathcal{A}(s_1, y) \Rightarrow \tilde{\xi}_{s_1}(s_2) > 0 \\ \phantom{s_1 \in A_2 \text{ and }} \Rightarrow \sin \frac{\theta(s_1) - \theta(s_2)}{2} < 0 \quad \Rightarrow \theta(s_2) > \theta(s_1) \text{ in } (-\frac{\pi}{2}, 0), \end{array} \right. $$

otherwise stated, when $s$ moves counter-clockwise on $A_1$, resp. $A_2$, the function $\theta(s)$ increases from 0 to $\frac{\pi}{2}$, resp. from $-\frac{\pi}{2}$ to 0, see Figure 9.10.
<figure><img src="image_36.png" /><figcaption>FIGURE 9.10. Sign of $\tilde{\xi}_{s_1}(s_2)$</figcaption></figure>

Under the assumption that $\dim V_{y,z(y)} = 2$, we are in a framework similar to that of Section 7.6, with $V_{y,z(y)}$ replacing $U_x^2$. We use a rotating function argument similar to the one used in Paragraph 7.6.4. For this purpose, we first investigate the limits of $\xi_s$ and $\theta(s)$ when $s$ tends to $z(y)$ or to $y$.

Let $\gamma_z$ (resp. $\gamma_y$) denote a local parametrization of $\Gamma$ in a neighborhood of $z(y)$ (resp. $y$), such that $\gamma_z(0) = z(y)$, $\gamma_z(-\varepsilon) \in A_2$, and $\gamma_z(\varepsilon) \in A_1$ (resp. $\gamma_y(0) = y$, $\gamma_y(-\varepsilon) \in A_2$, and $\gamma_y(\varepsilon) \in A_1$). Using our choice of sign for $\tilde{w}_y$ and $\tilde{v}_y$, the vanishing properties of these functions, and Lemma 3.14, we find that, in a pointed neighborhood of $z(y)$,

$$ egin{cases} 
\tilde{w}_y(\gamma_z(t)) = \alpha_w t + o(t), & \text{with } \alpha_w > 0, \\
\tilde{v}_y(\gamma_z(t)) = \alpha_v t^2 + o(t^2), & \text{with } \alpha_v > 0, \\
a(\gamma_z(t)) = \frac{\alpha_v}{\alpha_w} |t| + o(t), \\
b(\gamma_z(t)) = \operatorname{sgn}(t) + o(1). 
\end{cases} $$

Similarly, in a neighborhood of $y$,

$$ egin{cases} 
\tilde{w}_y(\gamma_y(t)) = \beta_w t^{2k-3} + o(t^{2k-3}), & \text{with } \beta_w > 0, \\
\tilde{v}_y(\gamma_y(t)) = \beta_v t^{2k-4} + o(t^{2k-4}), & \text{with } \beta_v > 0, \\
a(\gamma_y(t)) = 1 + o(1), \\
b(\gamma_y(t)) = \frac{\beta_w}{\beta_v} t + o(t). 
\end{cases} $$

This gives us the limits of $\xi_s$ and $\theta(s)$ when $s$ tends to $z(y)$ in $A_1$ or $A_2$ (resp. when $s$ tends to $y$ in $A_1$ and $A_2$),

$$ (9.15) \qquad \left\{ \begin{array}{ll} \lim_{s \to z(y)} \xi_s = -v_y, & \lim_{s \to z(y)} \theta(s) = \frac{\pi}{2}, \\ \lim_{s \to y} \xi_s = w_y, & \lim_{s \to y} \theta(s) = 0, \\ \lim_{s \to z(y)} \xi_s = v_y, & \lim_{s \to z(y)} \theta(s) = 0, \\ \lim_{s \to y} \xi_s = w_y, & \lim_{s \to y} \theta(s) = -\frac{\pi}{2}. \end{array} \right. $$

When $s$ moves counter-clockwise from $y$ to $z(y)$ on $A_1$, $\theta(s)$ increases from 0 to $\frac{\pi}{2}$; when $s$ moves counter-clockwise from $z(y)$ to $y$ on $A_2$, $\theta(s)$ increases from $-\frac{\pi}{2}$ to 0.
<figure><img src="image_37.png" /><figcaption>FIGURE 9.11. Lemma 9.8: limits of $\xi_s$ when $s$ tends to $y$ or $z(y)$</figcaption></figure>

Let $\sigma \mapsto \Gamma(\sigma)$ be a parametrization of $\Gamma$ such that $\sigma \in [-\frac{\pi}{2}, \frac{\pi}{2}]$, $\gamma_1(-\frac{\pi}{2}) = \gamma_1(\frac{\pi}{2}) = z(y)$, $\gamma_1(0) = y$, $\gamma_1((-\frac{\pi}{2}, 0)) = A_2$, and $\gamma_1((0, \frac{\pi}{2})) = A_1$.
Consider the map

$$ \theta_1 : (-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}) \ni \sigma \mapsto \theta(\gamma_1(\sigma)) \in (-\frac{\pi}{2}, \frac{\pi}{2}). $$

Then, $\theta_1$ extends to a continuous, increasing map from $(-\frac{\pi}{2}, \frac{\pi}{2})$ to $(-\frac{\pi}{2}, \frac{\pi}{2})$ such that $\lim_{\sigma \to \pm \frac{\pi}{2}} \theta_1(\sigma) = \pm \frac{\pi}{2}$ and $\lim_{\sigma \to 0} \theta_1(\sigma) = 0$.

For $t \in [-\frac{\pi}{2}, \frac{\pi}{2}]$, introduce the functions

$$ (9.16) \qquad \zeta_t := \cos t w_y - \sin t v_y, $$

$\zeta_{-\frac{\pi}{2}} = v_y$, $\zeta_0 = w_y$, and $\zeta_{\frac{\pi}{2}} = -v_y$. If $t \notin \{-\frac{\pi}{2}, 0, \frac{\pi}{2}\}$ there exists a unique $s(t) \in \Gamma\{y, z(y)\}$ such that

$$ (9.17) \qquad \left\{ \begin{array}{l} \rho(\zeta_t, y) = (2k-4), \quad \rho(\zeta_t, z(y)) = 1, \quad \rho(\zeta_t, s(t)) = 1, \\ S_b(\zeta_t) = \{y, z(y), s(t)\}, \\ s(t) \in A_1 \text{ if } t \in (0, \frac{\pi}{2}), \text{ and } s(t) \in A_2 \text{ if } t \in (-\frac{\pi}{2}, 0). \end{array} \right. $$

Indeed, near $z(y)$, $\cos t > 0$ implies that $\zeta_t$ has the sign of $w_y$. In a small pointed arc $J_y$ around $y$, $\zeta_t \sim -\sin t v_y$ which has the sign of $(-t)$.

We now apply the “rotating function” argument, see Paragraph 7.6.4, to the family of nodal sets $\mathcal{Z}(\zeta_t)$. We have $\mathcal{Z}(\zeta_{-\frac{\pi}{2}}) = \mathcal{Z}(\zeta_{\frac{\pi}{2}}) = \mathcal{Z}(v_y)$; when $t \in (0, \frac{\pi}{2})$, $S_b(\zeta_t) = \{y, z(y), s(t)\}$ with $s(t) \in A_1$; when $t \in (-\frac{\pi}{2}, 0)$, $s(t) \in A_2$.

Figure 9.12 (resp. Figure 9.13) illustrates the deformation of the nodal set $\mathcal{Z}(v_y)$ given in Subfigure (A) in a particular case with $k = 4$.

In Figure 9.12 the nodal set $\mathcal{Z}(v_y)$ is connected. When $t$ decreases from $\frac{\pi}{2}$ to 0 (top line), the nodal set $\mathcal{Z}(\zeta_t)$ deforms from $\mathcal{Z}(v_y)$ in Subfigure (A) to $\mathcal{Z}(w_y)$ in Subfigure (L). When $t$ increases from $-\frac{\pi}{2}$ to 0 (bottom line), the nodal set $\mathcal{Z}(\zeta_t)$ deforms from $\mathcal{Z}(v_y)$ in Subfigure (A) to $\mathcal{Z}(w_y)$ in Subfigure (R). In Figure 9.13 the nodal set $\mathcal{Z}(\zeta_t)$ has two connected components and deforms to (R) or (L).
<figure><img src="image_38.png" /><figcaption>FIGURE 9.12. The point $s$ tends to $y$ clockwise (top) or counter-clockwise (bottom), here $k = 4$</figcaption></figure>

<figure><img src="image_39.png" /><figcaption>FIGURE 9.13. The point $s$ tends to $y$ clockwise (top) or counter-clockwise (bottom), here $k = 4$</figcaption></figure>

The nodal patterns ($L$) and ($R$) belong to a function $w_y$. We claim that they are different. For this purpose, we label the loops as in Paragraph 7.6.2, and we use the combinatorial type of the function $w_y$, see Paragraph 8.2.2.

<figure><img src="image_40.png" /><figcaption>FIGURE 9.14. The nodal patterns ($L$) and ($R$) are different</figcaption></figure>
The maps $\tau$ describing the combinatorial types of the nodal patterns (L) and (R) of Figure 9.14 are given by

$$ \tau_L = \begin{pmatrix} \downarrow & 1 & 2 & 3 & 4 & 5 \\ 5 & 2 & 1 & 4 & 3 & \downarrow \end{pmatrix} \quad \text{and} \quad \tau_R = \begin{pmatrix} \downarrow & 1 & 2 & 3 & 4 & 5 \\ 1 & \downarrow & 3 & 2 & 5 & 4 \end{pmatrix}, $$

where $\downarrow$ corresponds to the arc hitting the boundary. Correspondingly, we label the nodal domains as in Paragraph 7.6.3, and we find the words,

$$ m_L = 121314 \quad \text{and} \quad m_R = 123242. $$

The criterion (7.21) given in Paragraph 7.6.3 shows that the nodal patterns (L) and (R) are different, although they should both be the nodal pattern of $w_y$, a contradiction. Recall that we already proved that $\dim V_{y,s} \le 2$. Since the assumption $\dim V_{y,s} = 2$ leads to a contradiction, at least in the example at hand, we conclude that $\dim V_{y,s} = 1$. The proof in the general case follows the same lines, as in Subsection 7.6. This proves Assertion (i).

**Remark 9.9.** When $t$ varies in $(-\frac{\pi}{2}, \frac{\pi}{2})$, the nodal sets $\mathcal{Z}(\zeta_t)$ vary continuously with respect to the Hausdorff distance, see Lemma 3.15. It follows that they are either all connected, or that they all have two connected components, so that their type in Figure 9.15 is either (b) & (c) or (d) & (e).

* Proof of Assertion (iii).* This is a consequence of Assertion (i) and its proof.

* Proof of Assertion (iv).* The fact that $\dim V_{y,s} = 1$ implies that the linear system which defines $v_{y,s}$ up to scaling has constant rank, so that it has a solution which depends smoothly on the parameters $y, s$ locally.

Lemma 9.8 is proved. $\square$

9.2.2. *Structure and combinatorial type of nodal sets in $V_{y,s}$. The last two lines in (9.3) give rise to three cases.*

**Case 1.** $\rho(u, y) = 2k-3$, $\rho(u, s) = 1$, and $S_b(u) = \{y, s\}$. This means that $u \in W_y$, and this case only occurs when $s = z(y)$, see Figure 9.15(a).

**Case 2.** $\rho(u, y) = 2k-4$, $\rho(u, s) = 2$, and $S_b(u) = \{y, s\}$, with two possibilities for $\mathcal{Z}(u)$,

* either $\mathcal{Z}(u)$ consists of $(k-2)$ loops at $y$ which do not intersect nor meet $\Gamma$ away from $y$, and one loop at $s$ which does not hit $\Gamma$ away from $s$, and does not meet the loops at $y$,

* or $\mathcal{Z}(u)$ consists of $(k-3)$ loops at $y$ which do not intersect nor meet $\Gamma$ away from $y$, and two simple arcs from $y$ to $s$ which do not meet except at $y$ and $s$, and do not meet the loops except at $y$; in this case we have a “generalized loop” which hits $\Gamma$ at $s$,

see Figures 9.15(b) and 9.15(d).

**Case 3.** $\rho(u, y) = 2k-4$, $\rho(u, s) = 1$, and there exists another point $s_1 \in \Gamma$, $s_1 \neq s, y$ such that $S_b(u) = \{y, s, s_1\}$ and $\rho(u, s_1) = 1$. In this case there are two possibilities for $\mathcal{Z}(u)$,

* either $\mathcal{Z}(u)$ consists of $(k-2)$ loops at $y$ which do not intersect nor meet $\Gamma$ away from $y$, and one arc from $s$ to $s_1$ which does not hit $\Gamma$ away from $s, s_1$, and does not meet the loops at $y$,
◇ or $\mathcal{Z}(u)$ consists of $(k-3)$ loops at $y$ which do not intersect nor meet $\Gamma$ away from $y$, and two simple arcs, one from $y$ to $s$ and one from $y$ to $s_1$ which do not meet except at $y$, and do not meet the loops except at $y$; in this case we have a “generalized loop” which contains a sub-arc of $\Gamma$ from $s$ to $s_1$, see Figures 9.15(c) and 9.15(e).

<figure><img src="image_41.png" /><figcaption>FIGURE 9.15. Nodal patterns for $u \in V_{y,s}$ ($k=4$)</figcaption></figure>

**Remarks 9.10.**

(i) The subcases in Cases 2 and 3 are distinguished by the fact that $b_0(\mathcal{Z}(u)) = 1$ (as in Figures 9.15(d) & (e)) or $b_0(\mathcal{Z}(u)) = 2$, as in Figures 9.15(b) & (d), if $s \neq z(y)$. The subcases cannot occur simultaneously. Indeed, by Lemma 3.12 we would otherwise find a function $u$ such that $\rho(u, y) \ge (2k-3)$ and $\rho(y, s) \ge 1$, with $s \neq z(y)$, contradicting Case 1.

(ii) At this stage of the discussion, the location of $z(y)$ with respect to $y$, $s$, and $s_1$ in Sub-figures 9.15(b)–(e) is not clear. This will be explained in Lemma 9.13.

For a pair $(y, s) \in \Gamma_{(2k-3)} \times \Gamma$ with $y \neq s$, and $\dim V_{y,s} = 1$, we can define the combinatorial type of a generator $v_{y,s}$ at the point $y$, as we did for the generator $w_y$ of $W_y$ in Paragraph 8.2.2, taking the above cases into account.

When $(y, s) = (y, z(y))$, the combinatorial type is that of $w_y$, and we denote it by $\tau_{w_y}$. For example, the combinatorial type $\tau_{w_y}$ of the function $w_y$ whose nodal pattern appears in Figure 9.15(a) is given by

$$ \tau_a^{9.15} = \begin{pmatrix} \downarrow & 1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & \downarrow & 5 & 4 \end{pmatrix}, $$
When $s \neq y$, the combinatorial type $\tau := \tau_{v_{y,s}}$ of a function $v_{y,s}$ is described as follows. When the nodal set is connected, as in Figure 9.15(d) and (e), we write $\tau(j) = s$ (resp. $s_1$) to indicate that the nodal semi-arc emanating from $y$ tangentially to the ray labeled $j$ ends up at $s$ (resp. $s_1$). When the nodal set has two connected components, as in Figure 9.15(b) and (c), we write $\tau(s) = s$ to indicate that there is a loop at $s$, and $\tau(s) = s_1$ to indicate that there is a nodal arc from $s$ to $s_1$. We describe the maps $\tau$ by $2 \times (2k-2)$ matrices. The first row enumerates the rays at $y$ and the rays at $s$ and $s_1$ (counter-clockwise). With this convention, the combinatorial type $\tau_{v_{y,s}}$ of a function $v_{y,s}$ whose nodal pattern appears in Figure 9.15(b)-(e), is given by one of the following formulas.

$$ \tau_{2,b}^{9.15} = \begin{pmatrix} 1 & 2 & 3 & 4 & s & s \\ 2 & 1 & 4 & 3 & s & s \end{pmatrix}, \quad \tau_{2,c}^{9.15} = \begin{pmatrix} 1 & 2 & 3 & 4 & s_1 & s \\ 2 & 1 & 4 & 3 & s & s_1 \end{pmatrix}, $$

and

$$ \tau_{1,d}^{9.15} = \begin{pmatrix} 1 & 2 & 3 & 4 & s & s \\ s & 3 & 2 & s & 1 & 4 \end{pmatrix}, \quad \tau_{1,e}^{9.15} = \begin{pmatrix} 1 & 2 & 3 & 4 & s_1 & s \\ s_1 & 3 & 2 & s & 1 & 4 \end{pmatrix}. $$

**9.2.3. Precise description of $V_{y,s}$**. In this paragraph we analyze the behavior of a generator $v_{y,s}$ of $V_{y,s}$ when $y$ and $s$ vary. More precisely, Lemma 9.11 describes the behavior of $v_{y,s}$ when $y \in \Gamma_{(2k-3)}$ is fixed and $s$ tends to $y$ or to $z(y)$, and the behavior when $s \in \Gamma_{(2k-3)}$ is fixed and $y$ tends to $s$. Lemma 9.13 describes the global behavior of $v_{y,s}$ for a given $y \in \Gamma_{(2k-3)}$.

**Lemma 9.11.** Let $U := U(\lambda_k)$ with $k \geq 3$. Assume that $\Omega$ is simply connected and $\dim U = (2k-2)$. Given $(y, s) \in \Gamma_{(2k-3)} \times \Gamma$, with $y \neq s$, recall that

$$ V_{y,s} := \{u \in U \mid \rho(u, y) \geq 2k-4 \text{ and } \rho(u, s) \geq 1\}. $$

The functions in the one-dimensional space $V_{y,s}$ have the following properties.

(i) When $s = z(y)$, the function $\check{v}_{y,z(y)} = \check{w}_y$ vanishes on $\partial\Omega$ precisely at the points $y$ and $z(y)$, and changes sign at these points. When $s \neq z(y)$ and $S_b(v_{y,s}) = \{y, s, s'\}$ with $s \neq s'$, $\rho(v_{y,s}, s) = \rho(v_{y,s}, s') = 1$, the function $\check{v}_{y,s}$ vanishes on $\partial\Omega$ precisely at the points $y, s$ and $s'$, does not change sign at $y$, and changes sign at $s$ and $s'$. When $s \neq z(y)$ and $S_b(v_{y,s}) = \{y, s\}$ with $\rho(v_{y,s}, s) = 2$, the function $\check{v}_{y,s}$ vanishes on $\partial\Omega$ precisely at the points $y$ and $s$, and does not change sign.

(ii) For fixed $y \in \Gamma_{(2k-3)}$, and $s$ close enough to $z(y)$, $S_b(v_{y,s}) = \{y, s, s'\}$, with $s' \neq s, y$, and $\rho(v_{y,s}, s) = 1$, $\rho(v_{y,s}, s') = 1$. Furthermore, when $s$ tends to $z(y)$, $[v_{y,s}]$ tends to $[w_y]$, and $s'$ tends to $y$.

(iii) For fixed $y \in \Gamma_{(2k-3)}$, and $s$ close enough to $y$, $S_b(v_{y,s}) = \{y, s, s'\}$, with $s' \neq s, y$, and $\rho(v_{y,s}, s) = 1$, $\rho(v_{y,s}, s') = 1$. Furthermore, when $s$ tends to $y$, $[v_{y,s}]$ tends to $[w_y]$, and $s'$ tends to $z(y)$.

(iv) For fixed $s \in \Gamma_{(2k-3)}$, and $y \in \Gamma_{(2k-3)}$ close enough to $s$, $S_b(v_{y,s}) = \{y, s, s'\}$, with $s' \neq y, s$, and $\rho(v_{y,s}, s) = 1$, $\rho(v_{y,s}, s') = 1$. Furthermore, when $y$ tends to $s$, $[v_{y,s}]$ tends to $[w_s]$, and $s'$ tends to $z(s)$.

**Proof.**

◇ *Proof of Assertion (i).* Use Lemma 3.14, and the description of the possible nodal patterns for $V_{y,s}$ which follows from Lemma 9.8.

◇ *Proof of Assertion (ii).* Assume that the first statement is not true. Then, we can find a sequence $\{s_n\}$ tending to $z(y)$, and a corresponding sequence $\{v_n := v_{y,s_n}\} \subset$
$\mathbb{S}(U)$ such that $\rho(v_n, y) = (2k-4)$, $\rho(v_n, s_n) = 2$, and $v_n$ tends to some $v \in \mathbb{S}(U)$. Since the convergence is uniform in $C^m$ for fixed $m \ge 0$, it follows that $\rho(v, y) \ge (2k-4)$ and $\rho(v, z(y)) \ge 2$, see Remark 3.6 (lower semi-continuity of $\rho$), and hence $v \in V_{y,z(y)}$. By Lemma 9.8(iii), we must have $[v] = [w_y]$, and we reach a contradiction since $\rho(w_y, z(y)) = 1$. This proves the first statement.

We now prove the second statement. Considering $\{v_n\} \subset \mathbb{S}(U)$ such that $S_b(v_n) = \{y, s_n, s'_n\}$ with $s_n \neq s'_n$ and $s_n$ tends to $z(y)$, we may also assume that $s'_n$ tends to some $s'$, and $v_n$ tends to $v$. Then, $\rho(v, y) \ge (2k-4)$, and $\rho(v, z(y)) \ge 1$, $\rho(v, s') \ge 1$. Lemma 9.8(iii) implies that $[v] = [w_y]$, and $\check{v}(s') = 0$ so that $s' \in \{y, z(y)\}$ (where $\check{v}$ is defined by (3.12)). Assuming that $s' = z(y)$, we would have $\rho(v, z(y)) = 2$, leading to a contradiction. Indeed, in that case, $s_n$ and $s'_n$ would both tend to $z(y)$, and the derivative $\partial_b \check{v}_n$ of the function $\check{v}_n$ along the boundary would vanish at some $t_n$ on the smallest arc between $s_n$ and $s'_n$, with $t_n$ tending to $z(y)$. Passing to the limit, we would have $\partial_b \check{w}_y(z(y)) = 0$, implying that $\rho(w_y, z(y)) \ge 2$. Assertion (ii) is proved.

◇ *Proof of Assertion (iii).* Assume that the first statement is false. Then, we can find a sequence $\{s_n\}$ tending to $y$, and a corresponding sequence $\{v_n := v_{y,s_n}\} \subset \mathbb{S}(U)$ such that $\rho(v_n, y) = (2k-4)$, $\rho(v_n, s_n) = 2$, and $v_n$ tends to some $v \in \mathbb{S}(U)$. Then, $\rho(v, y) \ge (2k-4)$. Applying the local structure theorem to $v$, and following the arguments in the proof of Lemma 9.2, Part (C), we may choose $r_1 > 0$ such that the neighborhood $D^+(r_1)$ of $y$ satisfies Properties (B-a)-(B-d) in the proof of Lemma 9.2. We may also assume that the sequence $\{s_n\}$ is contained in $D^+(r_1)$. The nodal set of $v_n$ consists of either $(k-2)$ loops at $y$ and a loop at $s_n$, or $(k-3)$ loops at $y$ and two arcs from $y$ to $s_n$, see Figure 9.15(b-d). As in the proof of Lemma 9.2, Part (C), for $r_0 < r_1$ small enough, each loop at $y$ must cross $C^+(r_0)$ at (at least) two distinct points; so does each loop at $s_n$, and each arc from $y$ to $s_n$. We then conclude that $\rho(v, y) \ge (2k-2)$. Euler's formula then implies that actually $\rho(v, y) = (2k-2)$. This is not possible since $y \in \Gamma_{(2k-3)}$. This proves the first statement in Assertion (iii).

We now prove the second statement. If $s_n$ tends to $y$, we have $S_b(v_n) = \{y, s_n, s'_n\}$, with $s_n \neq s'_n$. The preceding argument also shows that no subsequence of $s'_n$ can tend to $y$. We may then assume that $s_n$ tends to $y$ and $s'_n$ tends to some $s' \neq y$, with $v_n$ tending to some $v$. Then, $\rho(v, y) \ge (2k-4)$, and the previous argument shows that $\rho(v, y) \ge (2k-3)$, and $\rho(v, s') \ge 1$. Lemma 8.3 then shows that $s' = z(y)$, and that $[v] = [w_y]$. Assertion (iii) is proved.

◇ *Proof of Assertion (iv).* See Figure 9.16.

Assume that the first statement in false. Then, there exists a sequence $\{y_n\} \subset \Gamma_{(2k-3)}$ which tends to $s$, with a corresponding sequence $\{v_n := v_{y_n,s}\} \subset \mathbb{S}(U)$ tending to some $v \in \mathbb{S}(U)$, and such that $\rho(v_{y_n,s}, s) = 2$. The convergence of $v_n$ to $v$ being uniform in $C^{2k}$, we have $\rho(v, s) \ge (2k-4)$. Lemma 3.15 and Lemma 9.8(ii) applied to $v_n$ imply that $\mathcal{Z}(v) \cup \Gamma$ is connected. Applying Euler's formula to $v$, we conclude that $(2k-4) \le \sum_{z \in S_b(v)} \rho(v, z) \le (2k-2)$. Since the functions $\check{v}_n$ do not change sign on $\Gamma$, $\check{v}$ does not change sign either, so that $\rho(v, s) \ne (2k-3)$. Applying the local structure theorem to $v$ at $s$, and using the same proof as in Lemma 9.2, Part (C), (with the disc $D^+(s, r_0)$ and circle $C^+(s, r_0)$ centered at $s$), we infer that $\rho(v, s) = (2k-2)$. Indeed, the nodal sets $\mathcal{Z}(v_n)$ consist either is $(k-2)$ loops at $y_n$ (including a special loop touching $s$), or $(k-3)$ loops at $y_n$ and a loop at $s$. Since
$y_n$ tends to $s$, for $r_0$ small enough, these loops must intersect $C^+(s; r_0)$ at $(2k-2)$ distinct points, and we can conclude as in the proof of Lemma 9.2.

To prove the second statement, we can now choose a sequence $\{y_n\}$ such that $\rho(v_n, s) = 1$ for $n$ large enough, so that $S_b(v_n) = \{y_n, s, s'_n\}$, with $s'_n \neq s$. An argument similar to the previous one, shows that no subsequence of $\{s'_n\}$ can tend to $s$. Since $s \in \Gamma_{(2k-3)}$, the only remaining possibility is that $\rho(v, s) = (2k-3)$, and hence that $v \in W_s$. Assertion (iv) is proved.

The proof of Lemma 9.11 is complete. $\square$

<figure><img src="image_42.png" /><figcaption>FIGURE 9.16. Proof of Lemma 9.11(iv)</figcaption></figure>

We can enhance the previous lemma by the following properties.

**Properties 9.12.** Assume that $\Omega$ is simply connected, and that $\dim U = (2k-2)$. Given $(y, s) \in \Gamma_{(2k-3)} \times \Gamma$ with $y \neq s$, there exists $\varepsilon_0$ such that

(i) for all $s \in \mathcal{A}_c(z(y); \varepsilon_0) \cup \mathcal{A}_c(y; \varepsilon_0) \setminus \{y, z(y)\}$, $\rho(v_{y,s}, s) = 1$;

(ii) for all $\varepsilon < \varepsilon_0$, there exists $\eta > 0$ such that for all $s \in \mathcal{A}_c(z(y); \eta) \setminus \{z(y)\}$, $S_b(v_{y,s}) = \{y, s, s'\}$ with $s' \in \mathcal{A}(y; \varepsilon) \setminus \{y\}$;

(iii) for all $\varepsilon < \varepsilon_0$, there exists $\eta > 0$ such that for all $s \in \mathcal{A}_c(y; \eta) \setminus \{y\}$, $S_b(v_{y,s}) = \{y, s, s'\}$ with $s' \in \mathcal{A}_c(z(y); \varepsilon) \setminus \{z(y)\}$.

Let $s_1 \notin \{y, z(y)\}$. Assume that the function $v_{y,s_1}$ satisfies $\rho(v_{y,s_1}, s_1) = 1$, ie $S_b(v_{y,s_1}) = \{y, s_1, s'_1\}$, with $s'_1 \neq s_1$. Then,

(a) there exists $\varepsilon_1 > 0$, such that for all $s \in \mathcal{A}_c(s_1; \varepsilon)$, $\rho(v_{y,s}, s) = 1$;

(b) for all $\varepsilon > 0$, there exists $\eta < \varepsilon_1$ such that for all $s \in \mathcal{A}_c(s_1; \eta)$, $s' \in \mathcal{A}_c(s'_1; \varepsilon)$, i.e. the map $s \mapsto s'$ is continuous.

**Lemma 9.13.** Assume that $\Omega$ is simply connected, and that $\dim U = (2k-2)$. Let $y \in \Gamma_{(2k-3)}$. Then, the following properties hold.

(i) For any $s \in \mathcal{A}(y, z(y))$, the function $v_{y,s}$ satisfies $S_b(v_{y,s}) = \{y, s, s'\}$ with $s' \in \mathcal{A}(y, z(y))$, possibly with $s = s'$.

(ii) There exists a unique $s_1 \in \mathcal{A}(y, z(y))$ such that $v_{y,s_1}$ satisfies $\rho(v_{y,s_1}, s_1) = 2$.

(iii) For all $s \in \mathcal{A}(s_1, z(y))$, $S_b(v_{y,s}) = \{y, s, s'\}$ with $s' \in \mathcal{A}(y, s_1)$. Furthermore when $s$ moves counter-clockwise in $\mathcal{A}(s_1, z(y))$, $s'$ moves clockwise in $\mathcal{A}(y, s_1)$.
(iv) For all $s \in \mathcal{A}(y, s_1)$, $S_b(v_{y,s}) = \{y, s, s'\}$ with $s' \in \mathcal{A}(s_1, z(y))$. Furthermore when $s$ moves counter-clockwise in $\mathcal{A}(y, s_1)$, $s'$ moves clockwise in $\mathcal{A}(s_1, z(y))$. Similar statements hold for the arc $\mathcal{A}(z(y), y)$.

The statements in Lemma 9.13 are illustrated in Figure 9.17 (for the arc $\mathcal{A}(y, z(y))$). The corresponding nodal patterns appear in Figure 9.22.

<figure><img src="image_43.png" /><figcaption>FIGURE 9.17. Lemma 9.13: Assertions (ii) and (iii)</figcaption></figure>

*Proof.* Choose a generator $w_y$ of $W_y$ such that $\check{w}_y$ is positive in $\mathcal{A}(y, z(y))$, and negative in $\mathcal{A}(z(y), y)$.

*Assertion (i).* Assume that the assertion is false: there exists some $s_0 \in \mathcal{A}(y, z(y))$ such that $v_0 := v_{y,s_0}$ satisfies $S_b(v_0) = \{y, s_0, s'_0\}$, with $s'_0 \in \mathcal{A}(z(y), y)$. Since $s_0 \neq s'_0$, Lemma 9.11(i) implies that $\check{v}_0$ vanishes on $\partial\Omega$ only at the points $y, s_0$ and $s'_0$, does not change sign at $y$, and changes sign at $s_0$ and $s'_0$. Choose $v_0$ such that $\check{v}_0 < 0$ in $\mathcal{A}(s_0, s'_0)$. For $s \neq y, z(y)$, introduce the function

$$ \xi_s = a_0(s)w_y - b_0(s)v_0, $$

where

$$ \left\{ \begin{array}{l} a_0(s) = \check{v}_0(s) (\check{v}_0^2(s) + \check{w}_y^2(s))^{-\frac{1}{2}}, \\ b_0(s) = \check{w}_y(s) (\check{v}_0^2(s) + \check{w}_y^2(s))^{-\frac{1}{2}}. \end{array} \right. $$

Since $b_0(s) \neq 0$, $\rho(\xi_s, y) = 2k - 4$ and $\rho(\xi_s, s) \geq 1$, so that $\xi_s \in V_{y,s}$.

Choose $s \in \mathcal{A}(s_0, z(y))$. Since $\check{v}_0$ only changes sign at $s_0$ and $s'_0$, $\check{\xi}_s(s'_0) > 0$. By Lemma 3.14, at $y$ along $\partial\Omega$, $\check{w}_y$ vanishes at order $(2k - 3)$, while $\check{v}_0$ vanishes at order $(2k - 4)$. In a pointed neighborhood $J \setminus \{y\}$ of $y$ in $\partial\Omega$, we have that $\check{\xi}_s \sim -b_0(s)\check{v}_0 < 0$, and hence $\check{\xi}_s$ vanishes at some $s' \in \mathcal{A}(s'_0, y)$. It follows that $\xi_s \in V_{y,s}$ with $S_b(\xi_s) = \{y, s, s'\}$, $\rho(\xi_s, s) = 1$, $\rho(\xi_s, s') = 1$.

Similarly, choosing $t \in \mathcal{A}(y, s_0)$, we have $\check{\xi}_t(s'_0) < 0$ and $\check{\xi}_t(z(y)) > 0$, and $\check{\xi}_t$ vanishes at some $t' \in \mathcal{A}(z(y), s'_0)$.

Finally, we conclude as above that $\xi_t \in V_{y,t}$ with

$$ S_b(\xi_t) = \{y, t, t'\}, \rho(\xi_t, t) = 1, \rho(\xi_t, t') = 1. $$

These arguments can be visualized on Figure 9.18.

From the assumed existence of $s_0$, we conclude that, for all $s \in \mathcal{A}(y, z(y)), \xi_s \in V_{y,s}$ and $S_b(\xi_s) = \{y, s, s'\}$, with $s' \in \mathcal{A}(z(y), y)$, $\rho(\xi_s, s) = 1$, $\rho(\xi_s, s') = 1$. Because $s \in \mathcal{A}(s_0, z(y))$ implies that $s' \in \mathcal{A}(s'_0, y)$, with a parallel statement for $t$,
<figure><img src="image_44.png" /><figcaption>FIGURE 9.18. Proof of Lemma 9.13(i)</figcaption></figure>

the previous argument also shows that when the point $s$ moves counter-clockwise in $\mathcal{A}(y, z(y))$, the point $s'$ moves counter-clockwise in $\mathcal{A}(z(y), y)$. According to Lemma 9.11, Assertions (ii) and (iii), $\xi_s$ tends to $w_y$ when $s$ tends to $y$ or to $z(y)$ in $\mathcal{A}(y, z(y))$. Looking at the behaviour of the nodal sets, we reach a contradiction as Figure 9.19 shows. Assertion (i) is proved. ✓

<figure><img src="image_45.png" /><figcaption>FIGURE 9.19. Proof of Lemma 9.13(i)</figcaption></figure>

**Remark.** The previous arguments implicitly use the fact that the combinatorial type of $v_{y,s}$ does not change when $y$ is fixed and $s$ varies in $\Gamma\setminus\{y\}$ (the proof is similar
to the proof of Lemma 8.4(iv)), see Paragraph 9.2.2. Note also that the reasoning in Figure 9.19 is actually quite general, and only depends on the position of the arc from $y$ to $z(y)$ with respect to the loops.

Proof of Assertion (ii).

<figure><img src="image_46.png" /><figcaption>FIGURE 9.20. Proof of Lemma 9.13(ii)</figcaption></figure>

Let $\gamma_1 : [0, \ell_1] \to \Gamma$ be an arc-length parametrization of $\Gamma$, such that $\gamma_1(t)$ moves counter-clockwise, and $\gamma_1(0) = \gamma_1(\ell_1) = y$, $\gamma_1(\ell) = z(y)$.

According to Lemma 9.11(i), and to the above Assertion (i), taking $s \in \mathcal{A}(y, z(y))$ close to $y$, we have $S_b(v_{y,s}) = \{y, s, s'\}$ with $s' \in \mathcal{A}(y, z(y))$ close to $z(y)$. If $s = \gamma_1(t)$ and $s' = \gamma_1(t')$, for $t$ positive small enough, we have $t < t'$. Choose any such point $s_0 = \gamma_1(t_0)$ such that $v_0 = v_{y,s_0}$ satisfies $S_b(v_0) = \{y, s_0, s'_0\}$, with $s'_0 = \gamma_1(t'_0)$ and $t_0 < t'_0$. By Lemma 9.11 (i), the function $\check{v}_0$ vanishes precisely at the points $s_0$ and $s'_0$ and changes sign at these points. We choose it so that it is positive on $\mathcal{A}(s_0, s'_0)$. Define $\xi_s$ as in the proof of Assertion (i) (now with $s'_0 \in \mathcal{A}(y, z(y))$). Take $s \in \mathcal{A}(y, s_0)$, $s = \gamma_1(t)$ with $0 < t < t_0$. With our previous choice of signs for $\check{v}_y$, we find that $\check{\xi}_s(z(y)) > 0$ and $\check{\xi}_s(s'_0) < 0$, so that $\check{\xi}_s$ vanishes at some $s'$ in $\mathcal{A}(s'_0, z(y))$. Since $\xi_s \in V_{y,s}$, we have $S_b(\xi_s) = \{y, s, s'\}$, with $s \in \mathcal{A}(y, s_0)$, $s' \in \mathcal{A}(s'_0, z(y))$, $\rho(\xi_s, s) = \rho(\xi_s, s') = 1$. Then, $s' = \gamma_1(t')$ with $0 < t < t_0 < t'_0 < t' < \ell$, so that the map $(0, t_0) \ni t \mapsto t'$ is decreasing.

Introduce the set

$$J := \{t \in (0, \ell) \mid S_b(v_{y,\gamma_1(t)}) = \{y, \gamma_1(t), \gamma_1(t')\} \text{ with } t < t'\}.$$

If $t \in J$, $\rho(v_{y,\gamma_1(t)}, \gamma_1(t)) = \rho(v_{y,\gamma_1(t)}, \gamma_1(t')) = 1$. We have $(0, t_0) \subset J$ so that $J \neq \emptyset$, and hence $s_1 := \sup J$ exists. Since $\rho(v_{y,s}, s) = 1$ is an open condition, $s_1 \notin J$. Take a subsequence $\{t_n\} \subset J$ tending to $s_1$, and choose corresponding functions $v_{y,t_n} \in S(U) \cap V_{y,t_n}$. A subsequence converges to some function $v_1$ in $S(U)$ which satisfies $\rho(v_1, y) \ge (2k-4)$ and $\rho(v_1, s_1) \ge 1$. By Lemma 9.8, this implies that $v_1 \in V_{y,s_1}$ (use Remark 3.6). Since $s_1 \notin J$, we must have $\rho(v_1, s_1) = 2$, so that $v_1$ is a generator $v_{y,s_1}$ of $V_{y,s_1}$.

Proof of Assertions (iii) and (iv). Take $v_1 = v_{y,s_1}$ given by Assertion (ii). By Lemma 9.11 (i), we may choose this function such that $\check{v}_1 \ge 0$ and vanishes only at
y and $s_1$. For $s \neq y, z(y)$, introduce the functions,

$$ \xi_s = a_1(s)w_y - b_1(s)v_1, $$

where

$$ \left\{ \begin{array}{l} a_1(s) = \check{v}_1(s) (\check{v}_1^2(s) + \check{w}_y^2(s))^{-\frac{1}{2}}, \\ b_1(s) = \check{w}_y(s) (\check{v}_1^2(s) + \check{w}_y^2(s))^{-\frac{1}{2}}. \end{array} \right. $$

<figure><img src="image_47.png" /><figcaption>FIGURE 9.21. Proof of Lemma 9.13, Assertion (iii)</figcaption></figure>

When $s = s_1$, we have $\xi_{s_1} = -b_1(s_1)\check{v}_1 \le 0$, and $\xi_{s_1}$ only vanishes at $y$ and $s_1$; $\xi_{s_1} \in V_{y,s_1}$. When $s \in A(y, z(y)) \setminus \{s_1\}$, taking into account the properties of the functions involved, we find that

$$ \left\{ \begin{array}{l} \check{\xi}_s(z(y)) < 0, \\ \check{\xi}_s(s_1) > 0, \\ \check{\xi}_s < 0 \text{ in } J_y \setminus \{y\}, \end{array} \right. $$

where $J_y$ is a small arc on $\Gamma_1$ centered at $y$.

It follows that, for $s \in A(y, z(y)) \setminus \{s_1\}$, $\xi_s \in V_{y,s}$, $S_b(\xi_s) = \{y, s, s'\}$, $\rho(\xi_s, s) = \rho(\xi_s, s') = 1$, with the point $s$ on one side of $s_1$ in $A(y, z(y))$, and the point $s'$ on the other side. For these points $s, s'$, we have $V_{y,s} = V_{y,s'}$, so that $\xi_s$ and $\xi_{s'}$ must be proportional (Lemma 9.8 (i)), and since the functions $\xi_s$ and $\xi_{s'}$ both take a positive value at $s_1$, $\xi_{s'} = a\xi_s$, with $a > 0$.

We also have $\xi_s(t)\xi_{s'}(s) \le 0$, since

$$ \xi_s(t)\xi_{s'}(s) = -(\check{v}_1^2(s) + \check{w}_y^2(s))^{-\frac{1}{2}} (\check{v}_1^2(t) + \check{w}_y^2(t))^{-\frac{1}{2}} (\check{v}_1(s)\check{w}_y(t) - \check{w}_y(s)\check{v}_1(t))^2. $$

Choose $s \in A(s_1, z(y))$, and $t \in A(s, z(y))$. Then $s', t' \in A(y, s_1)$. The functions $\xi_s$ and $\xi_{s'}$ are positive at $s_1$, and hence positive respectively on the arcs $A(s', s)$ and $A(t', t)$. We have $\xi_s(t) < 0$, and hence, using the above properties, $\xi_{s'}(t) > 0$, and $\xi_{s'}(s') < 0$. This implies that $t' \in A(y, s')$. We have proved that when $s$ moves counter-clockwise in $A(s_1, z(y))$, $s'$ move clockwise in $A(y, s_1)$. This is coherent with Lemma 9.11(ii). The proof of Assertion (iii) is similar. $\square$
**Remarks 9.14.** (i) Lemma 9.13 corresponds to the first part of the proof of Lemma 3.5 in [HoMN1999] (from p. 1181, line (-7), “We consider the function” to p. 1182, line (+5), “the following nodal domain”).

Note however that, in Lemma 9.13, we do not assume that the point $z(y)$ is in the same connected component of $\Gamma_{(2k-3)}$ as the point $y$.

(ii) Figure 9.22 displays the possible nodal patterns of $w_y$, and the corresponding nodal patterns for the function $v_{y,s_1}$ with $s_1 \in \mathcal{A}(y, z(y))$ (see Lemma 9.13(ii)), and for the function $v_{y,s}$ with $s \in \mathcal{A}(s_1, z(y))$ (see Lemma 9.13(iii)).

<figure><img src="image_48.png" /><figcaption>FIGURE 9.22. Lemma 9.13 (ii) and (iii): nodal patterns for $w_y$ and $v_{y,s_1}$</figcaption></figure>

10. POSSIBLE STEPS TOWARDS A PROOF THAT $\mathrm{mult}(\lambda_k) \le (2k-3)$ FOR $k \ge 3$
Let $U := U(\lambda_k)$. throughout this section, we assume that

$$ (10.1) \qquad \begin{cases} \Omega \text{ is simply connected,} \\ k \ge 3 \text{ and } \dim U = (2k-2). \end{cases} $$

In [HoMN1999, Theorem B] the authors claim that under the above assumption, the bound $\mathrm{mult}(\lambda_k) \le (2k-3)$ holds for $k \ge 3$ (see [Berd2018, Section 4] for comments on the non simply connected case).

The purpose of this section is to revisit their scheme of proof, and point out some difficulties. We use the notation of the preceding sections.
10.1. **Preamble.** We fix a direct orthonormal frame {$\vec{e}_1, \vec{e}_2$} in $\mathbb{R}^2$, and the associated coordinates $\xi = (\xi_1, \xi_2)$. Without loss of generality, we may assume that the length of $\Gamma := \partial\Omega$ is equal to $2\pi$. We parametrize $\Gamma$ by arc-length, counter-clockwise, by some $\gamma_1: [0, 2\pi] \to \Gamma_1$, with unit tangent $\tau_1(t)$, and unit normal $\nu_1(t)$, so that the frame {$\tau_1(t), \nu_1(t)$} is direct and $\nu_1$ points inside $\Omega$. For $\varepsilon$ small enough, we define the curve $c(t, \varepsilon) = \gamma_1(t) + \varepsilon \nu_1(t)$.

We also fix {$\phi_1, \dots, \phi_{(2k-2)}$}, an orthonormal basis of the eigenspace $U$. According to Lemma 8.1, we have a $C^\infty$ map $x \mapsto [u_x]$ from (the interior of) $\Omega$ to $\mathbb{P}(U)$, where the function $u_x$ is a generator of the space $U_x$ defined by (8.3). The function $u_x$ is given by

$$ (10.2) \qquad u_x = \sum_{j=1}^{(2k-2)} \alpha_j(x) \phi_j $$

where the nonzero vector $\alpha(x) := (\alpha_1(x), \dots, \alpha_{(2k-2)}(x))$ is well defined up to scaling, given locally by a linear system, and $C^\infty$. Since $\Omega$ is simply connected, we can lift the map $x \mapsto \alpha(x)$ to a $C^\infty$ map from $\Omega$ to the unit sphere $S^{(2k-3)}$ or, equivalently, lift the map $x \mapsto [u_x]$ to a map $x \mapsto u_x$ from $\Omega$ to $\mathbb{S}(U)$. The main property of this map $x \mapsto u_x$ is that $\nu(u_x, x) = (2k-2)$ for all $x \in \Omega$. The least order homogeneous term in the Taylor expansion of the function $u_x$ near $x$ has degree $(k-1)$. More precisely, according to Bers's Theorem,

$$ (10.3) \qquad h_x(\xi) := \frac{d^{(k-1)}}{dt^{(k-1)}} u_x(x + t\xi) \Big|_{t=0} $$

is a nonzero, harmonic, homogeneous polynomial of degree $(k-1)$ in $\xi$. The expression (10.2) shows that the map $x \mapsto \bar{h}_x$ is $C^\infty$. Since $\bar{h}_x \neq 0$ for all $x$, we also obtain a map $\bar{h}$

$$ (10.4) \qquad \bar{h} : \begin{cases} x \mapsto \bar{h}_x \\ \Omega \to S_H \end{cases} $$

where $S_H$ is the unit circle in the 2-dimensional vector space $\mathcal{H}$ of harmonic, homogeneous polynomials of degree $(k-1)$ in $\mathbb{R}^2$.

Let $\gamma$ be any Jordan curve in $\Omega$. Restricting the map $\bar{h}$ to $\gamma$, we obtain a map $s \mapsto \bar{h}_{\gamma(s)}$ from $\gamma$ to $S_H$. This map has a degree $\deg(\bar{h}, \gamma)$. Since $\gamma$ is homotopic to a point, and since $\bar{h}$ is smooth, we conclude that $\deg(\bar{h}, \gamma) = 0$.

In order to show that the assumption (10.1) is absurd, hence that $\mathrm{mult}(\lambda_k) \le (2k-3)$ for any $k \ge 3$, it suffices to show that $\deg(\bar{h}, c(\cdot, \varepsilon)) \ne 0$ for $\varepsilon$ small enough. As we already pointed out in Remark 9.3, there is a difficulty here: the map $x \mapsto u_x$ extends to the boundary (see Lemma 9.2), and so does the map $x \mapsto h_x$. However, for $y \in \Gamma$, $\lim_{x \to y} h_x = 0$, and we have to look more closely at what happens near $\Gamma$, using Assertions (ii) and (iii) of Lemma 9.2.

The steps to achieve the previous goal are given in the next subsection, unfortunately with yet incomplete proofs.

10.2. **Steps for the proof that $\mathrm{mult}(\lambda_k) \le (2k-3)$ when $k \ge 3$.** The following two lemmas do not appear in [HoMN1999] and seem to be implicitly assumed.

**Lemma 10.1.** Assume that $\Omega$ is simply connected. Assume that $k \ge 3$ and that $\dim U = (2k-2)$. Then, $\Gamma_{(2k-2)}$ is not empty.
*Proof.* We are only able to give a partial proof.

**Claim 1.** Assume that $\Gamma_{(2k-3)} = \Gamma$. Then, $\inf_{y \in \Gamma} d(y, z(y)) > 0$. Indeed, the map $y \mapsto z(y)$ (see Lemma 8.4) is continuous on $\Gamma$ and $z(y) \neq y$. The result follows by compactness. Another argument is as follows. Assume that $\inf_{y \in \Gamma} d(y, z(y)) = 0$. Then, there exists a sequence $y_n$ such that $0 < d(y_n, z(y_n)) < 1/n$. To the sequence $\{y_n\}$ we associate a sequence of functions $\{w_n := w_{y_n}\} \in S(U)$ (Lemma 8.3). Up to taking subsequences, we may assume that $y_n$ converges to some $\bar{y}$, and that $w_n$ converges to some $w \in S(U)$. Since $y \mapsto z(y)$ is continuous, $z(y_n)$ converges to $z(\bar{y})$, and the assumption on the sequence implies that $z(y) = y$. By Lemma 9.2, $w \in W_{\bar{y}}$, and the arguments in the proof of this lemma imply that $\rho(w, \bar{y}) = (2k-2)$ which contradicts our assumption. ✓

**Claim 2.** Assume that $\Gamma_{(2k-3)} = \Gamma$. Then, there exists $\varepsilon_0 > 0$ such that for all $\varepsilon < \varepsilon_0$, and all $t \in [0, 2\pi]$, $S_b(u_{c(t,\varepsilon)}) \neq \emptyset$.

Assume this is not the case. Then, there exists a sequence $\{\varepsilon_n\}$ tending to zero, and a sequence $\{t_n\} \subset [0, 2\pi]$ such that $u_n := u_{c(t_n,\varepsilon_n)}$ satisfies $S_b(u_{c(t_n,\varepsilon_n)}) = \emptyset$. Without loss of generality, we may assume that $\{t_n\}$ tends to some $\bar{t}$. Then, $x_n := c(t_n, \varepsilon_n)$ tends to $y = c(\bar{t}, 0) \in \Gamma$. It follows that $[u_n]$ tends to $[w_y]$ and the proof of Lemma 9.2 implies that $\mathrm{ord}(w_y, y) = (2k-2)$, contradicting our assumption. ✓

**Claim 3.** Assume that $\Gamma_{(2k-3)} = \Gamma$. Then, there exists $\varepsilon_0 > 0$ and $\delta_0 > 0$ such that for all $\varepsilon < \varepsilon_0$, and all $t \in [0, 2\pi]$, $S_b(u_{c(t,\varepsilon)})$ has two elements whose distance on $\Gamma$ is bounded from below by $\delta_0$.

Assume this is not the case. From Claim 1, we already know that $S_b(u_{c(t,\varepsilon)}) \neq \emptyset$. Then there exists sequences $\{\varepsilon_n\}$ and $\{\delta_n\}$ tending to zero, and $\{t_n\}$ tending to some $\bar{t}$ such that $u_n := u_{c(t_n,\varepsilon_n)}$ has two boundary singular points $y_{n,1}$ and $y_{n,2}$ (possibly equal) whose distance is less than $\delta_n$. Taking subsequences, we may assume that these sequences tend to some $y \in \Gamma$, and the proof of Lemma 9.2 shows that $u_n$ tends to $w_y$ with $\rho(w_y, y) = (2k-2)$, a contradiction. ✓

Figure 10.1 displays the nodal sets $\mathcal{Z}(u_n)$ (blue) and $\mathcal{Z}(w_y)$ (red).

<figure><img src="image_49.png" /><figcaption>FIGURE 10.1. $\mathcal{Z}(u_n)$ (blue) and $\mathcal{Z}(w_y)$ (red)</figcaption></figure>

The assumption of the lemma implies that the combinatorial type of $w_y$ is constant along $\Gamma$, see Lemma 8.4, Assertion (iv). This means that the nodal arc from $y$ to $z(y)$ makes a constant angle $\theta$ with the tangent to $\Gamma$ at $y$.

Here is a *heuristic* argument to proceed. For $\varepsilon$ small enough, the nodal set $\mathcal{Z}(u_{c(t,\varepsilon)})$ contains a nodal arc which goes from $c(t, \varepsilon)$ to some $y_{c(t,\varepsilon),2}$ close to the point $z(\gamma_1(t))$.
Furthermore, this nodal arc is close to the arc from $\gamma_1(t)$ to $z(\gamma_1(t))$ (in the Hausdorff distance). As illustrated in Figure 10.1, its tangent at $c(t, \varepsilon)$ should make an angle $\theta_t$ with the tangent to $\Gamma$ at $\gamma_1(t)$, and $\theta_t$ should be close to $\theta$. For $\varepsilon$ small enough, the degree of the map $\bar{h}$ restricted to the curve $c(\cdot, \varepsilon)$ should therefore be one. A contradiction. We however do not know how to turn this argument into a rigorous proof of the lemma.

The (unproven) arguments in [HoMN1999] are as follows.

p. 1184, line (-2),

"We want to analyze how the star of tangents at $c(t, \varepsilon)$ to the $(k-1)$ nodal lines of $u_{c(t,\varepsilon)}$ crossing at $c(t, \varepsilon)$ turns if we follow $t$ from 0 to $2\pi$."

p. 1185, line (+1),

"To make this precise, we consider the continuous function $f: \Omega \to \mathbb{S}^1$ given by $f(x) = (2k-2)\alpha(x)$ modulo $2\pi$, where

$$t \mapsto t \exp \left( \frac{2\pi ik}{(2k-2)} + i\alpha(x) \right), \quad k = 0, \dots, (2k-3), \quad t \ge 0$$

are the tangents rays of the nodal lines through $x \in \mathcal{Z}(u_x)$. We want to analyze $f(c(t, \varepsilon))$.

p. 1185, line (+9),

"If $x \in \Omega$ is near enough to some point $y$ in the open set $\Gamma_{(2k-3)}$ then the nodal pattern $\mathcal{Z}(u_x)$ can be read off the nodal pattern $\mathcal{Z}(w_y)$, since the nodal domains move continuously. The nodal line leaving $\partial\Omega$ vertically stays connected to $\partial\Omega$ near $y$."

p. 1185, line (+22),

"This already implies that $f(c(t))$ follows the angle of $\partial D$ along each arc $(\partial D)_{2\ell-3}$."

□

**Lemma 10.2.** Assume that $\Omega$ is simply connected. Assume that $k \ge 3$ and that $\dim U = (2k-2)$. Then, $\Gamma_{(2k-2)}$ is not reduced to one point.

*Proof.* The proof would be clear if we knew that when $y \in \Gamma_{(2k-3)}$ tends to $\eta \in \Gamma_{(2k-2)}$, both $y$ and $z(y)$ are close to $\eta$, one on each side, compare with Remark 10.4. For the time being, we do not have a proof of this lemma. □

The next lemma appears as Lemma 3.5 in [HoMN1999]. It implicitly assumes that $\Gamma_{(2k-2)}$ contains at least two points.

**Lemma 10.3.** Assume that $\Omega$ is simply connected. Assume that $k \ge 3$ and that $\dim U = (2k-2)$. Let $C$ be a connected component of $\Gamma_{(2k-3)}$, and $y \in C$. Then, $z(y) \notin C$.

*Proof.* According to Lemmas 10.1 and 10.2, $\Gamma_{(2k-2)}$ has at least two elements.

Let $C := \mathcal{A}(\eta_1, \eta_2)$ be a connected component of $\Gamma_{(2k-3)}$, and $y \in \mathcal{A}(\eta_1, \eta_2)$. According to Lemma 9.13 (ii), there exists (a unique) $s_1 \in \mathcal{A}(y, z(y))$ such that the function $v_{y,s_1}$ satisfies $\rho(v_{y,s_1}, s_1) = 2$. Similarly, there is a unique $s_2$ in the arc $\mathcal{A}(z(y), y)$ such that $\rho(v_{y,s_1}, s_1) = 2$. If $z(y) \in \mathcal{A}(\eta_1, \eta_2)$, one of the points $s_1$ or $s_2$ also belongs to $\mathcal{A}(\eta_1, \eta_2)$.

Without loss of generality, we may start out with a point $y_1 \in C$, such that $z(y_1) \in C$, and there is a unique $s_1 \in \mathcal{A}(y_1, z(y_1))$ such that $\rho(v_{y_1,s_1}, s_1) = 2$. According
to Assertion (iv) of Lemma 9.11, for $y \in \mathcal{A}(y_1, s_1)$ close enough to $s_1$, we have $\rho(v_{y,s_1}, s_1) = 1$.

We now come back to the analysis of Lemma 3.5 in [HoMN1999] and consider the validity of the arguments given in [HoMN1999], starting from p. 1182, line (+6).

The first one claims (in our notation):

“Then we move $y \in \mathcal{A}(y_1, s_1)$ towards $s_1$ and consider $\mathcal{Z}(v_{y,s_1})$. One of the nodal arcs hitting at $s_1$ must move away before $y$ hits $s_1$ since there is no point of $\Gamma_{(2k-2)}$ in between,...”.

Assuming $y$ is close enough to $s_1$ so that $\rho(v_{y,s_1}, s_1) = 1$, applying Lemma 9.11, Assertion (iv), we have $S_b(v_{y,s_1}) = \{y, s_1, s'(y)\}$, and there are two possibilities, either $s'(y) \in \mathcal{A}(y, s_1)$ or $s'(y) \notin \mathcal{A}(y, s_1)$. (We write $s'(y)$ to mention the dependence with respect to $y$ ($s_1$ being fixed).) This claim is correct.

Then, [HoMN1999] claims that only the first case can occur for $y$ close enough to $s_1$, see p. 1182, line (+8),

“and it must move eventually towards $y$ [...] since the nodal type of $w_y$ [in our notation] is constant in $C$ [the connected component]”.

This claim is questionable but let us continue the proof (see below the discussion of the second possibility).

**First possibility.** Assuming that the first possibility actually holds, namely that for $y$ close enough to $s_1$, $s'(y) \in \mathcal{A}(y, s_1)$. Letting $y$ tend to $s_1$, we find some $y_2 \in \mathcal{A}(y_1, z(y_1))$ such that $s'(y_2) = y_2$ which implies that $\rho(v_{y_2,s_1}, y_2) = 2k - 3$, hence $z(y_2) = s_1$, see Figure 10.2. Summing up, we have $y_2, z(y_2) \in \mathcal{A}(y_1, z(y_1))$ and, measuring distances on the arc $\mathcal{A}(y_1, z(y_1))$, $d(y_2, z(y_2)) < d(y_1, z(y_1))$.

To end up the argument in [HoMN1999], p. 1182, line (+10),

“We have then the same situation ... a contradiction.”, we can consider the set

$$K := \{y \in \mathcal{A}[y_1, z(y_1)] \mid z(y) \in \mathcal{A}[y_1, z(y_1)]\}$$

and the number $d := \inf\{d(y, z(y)) \mid y \in K\}$. Since the set $K$ is non-empty and closed, the infimum is achieved. The previous argument shows that $d = 0$, and hence that there exists some $y_0$ in $K$ such that $z(y_0) = y_0$ which contradicts the fact that $y_0 \in \Gamma_{(2k-3)}$. So finally, if $y \in \Gamma_{(2k-3)}$, $y$ and $z(y)$ cannot belong to the same connected component of $\Gamma_{(2k-3)}$. This would end the proof of Lemma 10.3, i.e., the proof of [HoMN1999], Lemma 3.5, assuming that at each step only the first possibility occurs.

<figure><img src="image_50.png" /><figcaption>FIGURE 10.2. 1st possibility</figcaption></figure>
**Second possibility.** [HoMN1999] seems to claim that this possibility leads to an
immediate contradiction and refers to the nodal type of $w_y$. Figure 10.3 seems to
invalidate this claim. As far as we understand, Lemma 10.3 remains unproven. □

<figure><img src="image_51.png" /><figcaption>FIGURE 10.3. 2nd possibility: apparent counterexample to the claim on the nodal type</figcaption></figure>

**Remark 10.4.** It might be sufficient to prove the following weaker statement: if $C = A(\eta_1, \eta_2)$, when $y$ is close enough to $\eta_i$, the point $z(y)$ is not in $C$ (so that $y$ and $z(y)$ are not on the same side of $\eta_i$).

Lemma 3.6 in [HoMN1999] contains the following assertions.

(i) If $y$ moves clockwise to $\eta_1$ then $z(y)$ moves anticlockwise to $\eta_1$.

(ii) If $y$ moves anti-clockwise to $\eta_2$ then $z(y)$ moves clockwise to $\eta_2$.

(iii) In particular the nodal pattern of $Z(w_{\eta_1})$ and $Z(w_{\eta_2})$ are different.

(iv) Moreover $\Gamma_{(2k-3)}$ consists of only finitely many open arcs and $\Gamma_{(2k-2)}$ is a finite set of $\Gamma$.

<figure><img src="image_52.png" /><figcaption>FIGURE 10.4. $Z(w_{\eta_1})$ and $Z(w_{\eta_2})$ have different combinatorial types</figcaption></figure>

Some comments.

◇ If *clockwise* and *anticlockwise* are meant to indicate some monotonic behavior, Assertions (i) and (ii) are not proved in [HoMN1999]. It is not clear whether knowing this precise behavior of the map $y \mapsto z(y)$ is necessary.
* Assuming Lemma 10.3 (their Lemma 3.5) is true, and taking Lemma 8.4(iii) into account, we see that when $y \in \Gamma_{(2k-3)}$ tends to some $\eta \in \Gamma_{(2k-2)}$ the points $y$ and $z(y)$ do not lie on the same side of $\eta$. According to Remark 10.4, this property seems sufficient to prove that the eigenfunctions $w_{\eta_1}$ and $w_{\eta_2}$ have different combinatorial types (same proof as in Subsection 7.6), giving a precise meaning to Assertion (iii). This is illustrated in Figure 10.4. This fact does not seem to be used in the sequel.

* More interestingly, assuming that when $y \in \Gamma_{(2k-3)}$ tends to $\eta \in \Gamma_{(2k-3)}$ the points $y$ and $z(y)$ do not lie on the same side of $\eta$, we see that the combinatorial type of $w_y$ changes when $y$ passes through some $\eta \in \Gamma_{(2k-2)}$. This is illustrated in Figure 10.5.

* Assertion (iv) is proved in our Lemma 8.4(ii).

<figure><img src="image_53.png" /><figcaption>FIGURE 10.5. The combinatorial types of $Z(w_y)$ are different on the right and on the left of $\eta \in \Gamma_{(2k-2)}$</figcaption></figure>

The last argument in the proof should be as follows.

**Lemma 10.5.** For $\varepsilon$ small enough, the smooth mapping $t \mapsto \bar{h}_{c(t,\varepsilon)}$ from the circle $c(\cdot, \varepsilon)$ to $S_H$ has nonzero degree.

*Proof.* We do not know how to prove this lemma.

The argument given in [HoMN1999, p. 1186] is:

p. 1186, line (-5),

> "So finally, the smooth mapping $t \mapsto f(c(t, \varepsilon))$, $S^1 \to S^1$ has mapping degree $2\pi\#(\partial D)_{2\ell-2} + 2\pi(2\ell - 2) > 0$ and cannot be null homotopic. But by construction it is continuously extended into the interior of the circle and this is null homotopic, a contradiction."

Note that we have the same difficulty as in the proof of Lemma 10.1. □

<figure><img src="image_54.png" /></figure>

**Concluding remarks.** Unless we modify the strategy of proof, we are led to two main questions.

**Question 1.** Given $\eta \in \Gamma_{(2k-2)}$ and $y \in \Gamma \setminus \{\eta\}$ close to $\eta$, $z(y)$ is close to $\eta$ as well, so that $y$ and $z(y)$ are both in a small arc $A$ centered at $\eta$. Is it true that $y$ and $z(y)$
are not on the same connected component of $A\setminus\{\eta\}$ (i.e., that they are on either sides of $\eta$)?

If so, we could prove that the combinatorial type of $w_y$ (which is constant in each connected component of $\Gamma_{(2k-3)}$) changes each time $y$ crosses some $\eta \in \Gamma_{(2k-2)}$. This would in particular prove Lemma 10.2.

One way to prove Lemma 10.1 and Lemma 10.5 would be to better understand how the *star* at $x$, i.e. the zero set of the harmonic polynomial $h_x$, changes (rotates) when $x \in \Omega$ tends to $y \in \Gamma_{(2k-3)}$. If $x_n$ tends to $y$, the stars at $x_n$ have limit points because they can be identified to sets of equidistant points on the unit circle. This leads us to ask the following question.

*Question 2.* Does the star at $x$ have a (unique) limit when $x$ tends to $y \in \Gamma_{(2k-3)}$, and can we identify one branch of the limit star as the ray at $y$ corresponding to the nodal arc from $y$ to $z(y)$ ?

One difficulty here is that Hausdorff convergence of nodal sets is not sufficient.
## APPENDIX A. A REMARK ON UPPER BOUNDS FOR MULTIPLICITIES VS COURANT-SHARP EIGENVALUES

For simplicity, let us only consider Dirichlet eigenvalues in a smooth bounded domain $\Omega \subset \mathbb{R}^2$. The upper bounds on the eigenvalue multiplicities strongly rely on Courant's nodal domain theorem. As observed in [HoMN1999], they are actually a consequence of the following 3-step result, $n \in \{1, 2, 3\}$, provided that the third step ($n = 3$) is correct, see the final paragraph in Section 7.

**Proposition A.1.** Let $U(\lambda)$ is a Dirichlet eigenspace. Assume that

$$ \sup \{\kappa(u) \mid u \in U(\lambda)\} = \ell $$

for some $\ell \ge 3$. Then $\dim(U(\lambda)) \le (2\ell - n)$.

Indeed, since $u \in U(\lambda_k)$ implies that $\kappa(u) \le k$:

(1) The first step, $n = 1$, yields the upper bound $\mathrm{mult}(\lambda_k \le (2k-1))$ (Theorem 1 in [Nadi1987]).

(2) The second step, $n = 2$, yields the upper bound $\mathrm{mult}(\lambda_k) \le (2k - 2)$ (Lemma 2.13 in [HoMN1999] or Proposition 7.17).

(3) The third step, $n = 3$, yields the upper bound $\mathrm{mult}(\lambda_k) \le (2k - 3)$ (Theorem B in [HoMN1999]).

According to Pleijel [Plej1956], the equality in Courant's theorem can only occur for finitely many eigenvalues, the so-called Courant-sharp eigenvalues. If $\lambda_k$ is not a Courant-sharp eigenvalue, in particular if $k$ is large enough (depending on the geometry of the domain), $u \in U(\lambda_k)$ implies that $\kappa(u) \le (k-1)$, and the above proposition implies that $\mathrm{mult}(\lambda_k) \le (2k-2-n)$. When $\lambda_k$ is not Courant-sharp, the upper bound for the multiplicity is improved by 2.

The above proposition can be restated as

$$ (A.1) \qquad \mathrm{mult}(\lambda_k) \le 2 \sup_{u \in U(\lambda_k)} \kappa(u) - 1. $$

Note that this implies that

$$ \limsup_{k \to +\infty} \frac{\mathrm{mult}(\lambda_k)}{k} \le 2. $$

We can continue the discussion a little further recalling Pleijel's proof. Let $u$ be an eigenfunction in $U(\lambda_k)$ with precisely $\kappa(\lambda_k) := \sup_{u \in U(\lambda_k)} \kappa(u)$ nodal domains. Applying the Faber-Krahn inequality to each nodal domain, we obtain

$$ \lambda_k |\Omega| \ge \kappa(\lambda_k) \pi j_{0,1}^2, $$

where $|\Omega|$ denotes the area of $\Omega$. Then,

$$ (A.2) \qquad \mathrm{mult}(\lambda_k) \le \frac{2}{\pi j_{0,1}^2} \lambda_k |\Omega| - 1. $$

For a regular bounded domain $\Omega$ in $\mathbb{R}^2$, Weyl's asymptotic formula reads

$$ (A.3) \qquad N(\lambda) := \#\{j \mid \lambda_j < \lambda\} = \frac{|\Omega|}{4\pi} \lambda + O(\sqrt{\lambda}). $$

**Remark A.2.** For this formula, we refer to Ivrii's papers [Ivri1980r, Ivri1980], and to Hörmander's book [Horm2009d, Theorem 29.3.3]. These authors actually give a much more precise formula which yields a two-term asymptotic formula under
an assumption on the set of periodic billiard trajectories (see Corollary 29.3.4 in Hörmander's book).

It follows from (A.3) that

$$ (A.4) \qquad \limsup_{k \to +\infty} \frac{\kappa(\lambda_k)}{k} \le \frac{|\Omega|}{\pi j_{0,1}^2} \lim_{k \to +\infty} \frac{\lambda_k}{k} = \frac{4}{j_{0,1}^2} =: \gamma < 1. $$

As a consequence, we obtain the better estimate (see [Plej1956])

$$ (A.5) \qquad \limsup_{k \to +\infty} \frac{\mathrm{mult}(\lambda_k)}{k} \le 2\gamma < 2. $$

On the other hand, for any $\delta$ small enough, $\mathrm{mult}(\lambda_k) = N(\lambda_k + \delta) - N(\lambda_k - \delta)$. By Weyl's asymptotic formula, we should have

$$ \mathrm{mult}(\lambda_k) = O(\sqrt{\lambda_k}), $$

and hence

$$ \limsup_{k \to +\infty} \frac{\mathrm{mult}(\lambda_k)}{k} = 0, $$

which shows that the inequality (A.5) is not pertinent.

For a bounded domain $\Omega \subset \mathbb{R}^n$ with $C^2$ boundary, and Dirichlet boundary condition, one can show that there exists a constant $C(\Omega)$, depending on $\Omega$ and invariant under dilations, such that for $k > C(\Omega)$, the $k$th Dirichlet eigenvalue $\lambda_k$ is not Courant-sharp. In dimension 2, the constant $C(\Omega)$ can be estimated in terms of the area $|\Omega|$, the length $|\partial\Omega|$ of the boundary, the curvature and the cut-distance of $\partial\Omega$. We refer to [BeHe2016, Theorem 1.3] for more details, and to [BeGi2016, Theorem 1] for an extension to less regular domains. The proof of this result makes use of a lower bound on the remainder term $R(\lambda) := N(\lambda) - \frac{|\Omega|}{4\pi} \lambda$ in Weyl's asymptotic estimate, as given for example in [BeLi2001].

**Remark A.3.** For the case of domains with Neumann or Robin boundary condition, we refer to [GiLe2020, Proposition 1.1]. This paper also considers the special case of convex domains.

**Remark A.4.** As in Bérard-Helffer [BeHe2016] (see also [BeGi2016]) we can think of a geometrical control of the constants (see Safarov [Safa2001] or Van den Berg-Lianantonakis [BeLi2001]) who give estimates in the form

$$ |N(\lambda) - C_n |\Omega|^{n/2}| \le C_{\mathrm{geom}}(\Omega) \lambda^{(n-1)/2} \ln \lambda. $$

Although we only use lower bounds of the remainder in the proof of Pleijel's theorem, we will also need here upper bounds.

# APPENDIX B. THE LOCAL STRUCTURE OF THE NODAL SET IN THE NEIGHBORHOOD OF A CRITICAL ZERO

In this Appendix, we provide a proof of the local structure theorem for the nodal set of an eigenfunction $u$ in the neighborhood of an interior singular point $x$. Cheng [Chen1976] proposes a more precise result (existence of a local diffeomorphism which sends the nodal set of $u$ in a neighborhood of $x$ onto the nodal set of the higher order term in the Taylor expansion of $u$ at $x$ (a harmonic polynomial), which consists of rays. Our proof has the advantage of being more quantitative in particular when applied to eigenfunctions depending on a parameter. This proof is probably known
although we did not find any reference except the analysis in § 128 of the book
*Georges Valiron, Cours d'analyse mathématique. Théorie des fonctions. Masson,*
*1966.* cited in [Bess1980].

Let $(M, g)$ be a compact Riemannian surface, and let $V : M \to \mathbb{R}$ be a real valued smooth potential. Let $u \neq 0$ be a real valued function, satisfying

$$
(B.1) \qquad (-\Delta_g + V)u = \lambda u
$$

for some real number $\lambda$. Here $\Delta_g$ denotes the Laplace-Beltrami operator of $(M, g)$.
Let $x$ be a given (interior) point of $M$ (in case $M$ has a boundary).

Choose some $r_0 > 0$ such that the exponential map $\exp_x : T_x M \to M$ is a diffeomor-
phism from the disk $D(2r_0)$, with center 0 and radius $2r_0$ in $T_x M$, onto the geodesic
disk $D(x, 2r_0)$, with center $x$ and radius $2r_0$ in $M$. Choose an orthonormal frame in
$T_x M$, call $(\xi_1, \xi_2)$ the corresponding coordinates in $T_x M$, and $(r, \omega)$ the associated
polar coordinates. In the normal coordinates $(\xi_1, \xi_2)$, the Riemannian metric is given
by the $2 \times 2$ matrix $G = (g_{ij})$, where $g_{ij} = g(\frac{\partial}{\partial \xi_i}, \frac{\partial}{\partial \xi_j})$; the Riemannian measure is
given by $v_g d\xi_1 d\xi_2$, where $v_g = \sqrt{\det G}$. Write the matrix $G^{-1}$ as $G^{-1} = (g^{ij})$. Then
(see for example [GaHuLa2004, § 2.89bis, p. 87]),

$$
(B.2) \qquad \left\{
\begin{array}{ll}
G(0,0) = \mathrm{Id}, & \text{i.e. } g_{ij}(0,0) = \delta_{ij}, \quad 1 \le i,j \le 2, \\
\frac{\partial G}{\partial \xi_k}(0,0) = 0, & \text{i.e. } \frac{\partial g_{ij}}{\partial \xi_k}(0,0) = 0, \quad 1 \le i,j,k \le 2.
\end{array}
\right.
$$

It follows that

$$
(B.3) \qquad \left\{
\begin{array}{l}
v_g(0,0) = 1, \\
G^{-1}(0,0) = \mathrm{Id}, \quad \text{i.e. } g^{ij}(0,0) = \delta_{ij}, \quad 1 \le i,j \le 2, \\
\frac{\partial v_g}{\partial \xi_k}(0,0) = 0, \quad 1 \le k \le 2. \\
\frac{\partial G^{-1}}{\partial \xi_k}(0,0) = 0, \quad \text{i.e. } \frac{\partial g^{ij}}{\partial \xi_k}(0,0) = 0, \quad 1 \le i,j,k \le 2.
\end{array}
\right.
$$

Given a function $u$ on $M$, let $f = u \circ \exp_x$. In the local coordinates $(\xi_1, \xi_2)$, the
Laplace-Beltrami operator $\Delta_g$ is given (see [BeGM1971, §G.III, p. 126]) by

$$
(B.4) \quad \left\{
\begin{array}{@{}l@{}}
\Delta_g f = v_g^{-1} \sum_{1 \le i,j \le 2} \frac{\partial}{\partial \xi_i} (v_g g^{ij} \frac{\partial f}{\partial \xi_j}), \\
\phantom{\Delta_g f =} = \sum_{1 \le i,j \le 2} g^{ij} \frac{\partial^2 f}{\partial \xi_i \partial \xi_j} + \sum_{1 \le j \le 2} b_j \frac{\partial f}{\partial \xi_j}, \\
\text{where } b_j = \sum_{1 \le i \le 2} v_g^{-1} \frac{\partial}{\partial \xi_i} (v_g g^{ij}), \quad 1 \le j \le 2.
\end{array}
\right.
$$

Letting $\Delta_0 = \sum_{1 \le j \le 2} \frac{\partial^2}{\partial \xi_j^2}$ denote the Laplacian in the Euclidean space $(T_x M, g_x)$,
and taking relations (B.2) and (B.3) into account, we obtain the following expression
for the Laplace-Beltrami operator,

$$
(B.5) \qquad \left\{
\begin{array}{l}
\Delta_g = \Delta_0 + \sum_{1 \le i,j \le 2} a_{ij} \frac{\partial^2}{\partial \xi_i \partial \xi_j} + \sum_{1 \le j \le 2} b_j \frac{\partial}{\partial \xi_j}, \\
\text{where } \mathrm{ord}(a_{ij}, (0,0)) = 2 \text{ and } \mathrm{ord}(b_j, (0,0))) = 1.
\end{array}
\right.
$$

If $u$ satisfies (B.1) and $u(x) = 0$, the unique continuation theorem [Aron1957, DoFe1990a] implies that $f$ does not vanish at infinite order at 0. If $\mathrm{ord}(u, x) = \mathrm{ord}(f, 0) = p$, Taylor's formula at 0, gives

$$
(B.6) \quad \left\{
\begin{array}{@{}l@{}}
f(\xi_1, \xi_2) = \sum_{|\alpha|=p} \frac{1}{\alpha!} D^\alpha f(0,0) (\xi_1, \xi_2)^\alpha + R_{p+1}(\xi_1, \xi_2), \text{ where} \\
R_{p+1}(\xi_1, \xi_2) = \sum_{|\alpha|=p+1} \frac{p+1}{\alpha!} (\xi_1, \xi_2)^\alpha \int_0^1 (1-t)^p D^\alpha f(t\xi_1, t\xi_2) \, dt.
\end{array}
\right.
$$
Here, as usual,

$$
(B.7) \qquad \left\{
\begin{array}{l}
\alpha = (\alpha_1, \alpha_2), \quad |\alpha| = \alpha_1 + \alpha_2, \quad (\xi_1, \xi_2)^\alpha = \xi_1^{\alpha_1} \xi_2^{\alpha_2}, \text{ and} \\
D^\alpha f = \frac{\partial^{|\alpha|} f}{\partial \xi_1^{\alpha_1} \partial \xi_2^{\alpha_2}}.
\end{array}
\right.
$$

Using relations (B.1) and (B.5), and identifying the terms with lowest order, we find that the polynomial $P_p(\xi_1, \xi_2) := \sum_{|\alpha|=p} \frac{1}{\alpha!} D^\alpha f(0,0) (\xi_1, \xi_2)^\alpha$ is homogeneous of degree $p$, and harmonic with respect to $\Delta_0$.

**Remark B.1.** As a matter of fact, we may write a 2-term Taylor formula for the function $f$,

$$
f(\xi_1, \xi_2) = P_p(\xi_1, \xi_2) + P_{p+1}(\xi_1, \xi_2) + R_{p+1}(\xi_1, \xi_2),
$$

where $P_p$ and $P_{p+1}$ are homogeneous polynomials of degrees $p$ and $(p+1)$ respectively, and where the remainder term $R_{p+2}$ vanishes at order at least $(p+2)$. Then, we actually have that $\Delta_0 P_p = 0$ and $\Delta_0 P_{p+1} = 0$.

Writing the harmonicity condition $\Delta_0 P_p = 0$ in polar coordinates $(r, \omega)$ in $T_x M$, we find that the polynomial $P_p$ has the form

$$
(B.8) \qquad P_p(r \cos \omega, r \sin \omega) = a r^p \sin(p\omega - \omega_0).
$$

for some $0 \neq a \in \mathbb{R}$ and some $\omega_0 \in [0, 2\pi]$.

Multiplying the function $f$ by some constant, and rotating the coordinates $(\xi_1, \xi_2)$ in $\mathbb{R}^2$ if necessary, we can assume that $a = 1$ and $\omega_0 = 0$. It follows that $f$ can be written as

$$
(B.9) \qquad f(r \cos \omega, r \sin \omega) = r^p \sin(p\omega) + R_{p+1}(r \cos \omega, r \sin \omega),
$$

and we can write the second term as

$$
(B.10) \quad \left\{
\begin{array}{l}
R_{p+1}(r \cos \omega, r \sin \omega) = r^{p+1} T_{p+1}(r, \omega), \quad \text{where } T_{p+1}(r, \omega) \text{ is given by} \\
\sum_{|\alpha|=p+1} \frac{p+1}{\alpha!} (\cos \omega, \sin \omega)^\alpha \int_0^1 (1-t)^p D^\alpha f(tr \cos \omega, tr \sin \omega) dt.
\end{array}
\right.
$$

Define

$$
(B.11) \qquad W(r, \omega) := \sin(p\omega) + r T_{p+1}(r, \omega).
$$

The function $W(0, \omega)$ vanishes precisely for the values

$$
(B.12) \qquad \omega_j := j \frac{\pi}{p}, \quad j \in \{0, \dots, 2p-1\}.
$$

Choose $\alpha_1 \in (0, \frac{\pi}{8})$ and define $\alpha_p := \frac{\alpha_1}{p}$. We have the following relations.

$$
(B.13) \qquad \left\{
\begin{array}{l}
\sin(p(\omega_j \pm \alpha_p)) = \pm(-1)^j \sin(\alpha_1), \\
|\sin(p\omega)| \ge \sin\alpha_1, \text{ for } \omega \notin \bigcup_{j=0}^{2p-1} (\omega_j - \alpha_p, \omega_j + \alpha_p)
\end{array}
\right.
$$

Define

$$
(B.14) \quad r_1 := \min \left\{ r_0, \frac{1}{2} \sin(\alpha_1) \|T_{p+1}\|_{\infty, D(r_0)}^{-1}, \frac{1}{2} \cos(\alpha_1) \|\partial_\omega T_{p+1}\|_{\infty, D(r_0)}^{-1} \right\},
$$

where $\| \cdot \|_{\infty, D(\frac{1}{2})}$ denotes the $L^\infty$ norm of functions on the disk $D(r_0)$ of radius $r_0$ in $TxM$.

**Proposition B.2.** For any $r \le r_1$,
(i) the function $\omega \mapsto W(r, \omega)$ does not vanish in

$$ [0, 2\pi] \setminus \bigcup_{j=0}^{2p-1} (\omega_j - \alpha_p, \omega_j + \alpha_p) = \bigcup_{j=0}^{2p-1} [\omega_j + \alpha_p, \omega_{j+1} - \alpha_p]; $$

(ii) for each $j \in \{0, \dots, 2p-1\}$, the function $\omega \mapsto W(r, \omega)$ has exactly one zero $\tilde{\omega}_j(r) \in (\omega_j - \alpha_p, \omega + \alpha_p)$;

(iii) for each $j \in \{0, \dots, 2p-1\}$, the function $r \mapsto \tilde{\omega}_j(r)$ is $C^\infty$ in $(0, r_1)$ and tends to $\omega_j$ as $\rho$ tends to zero;

(iv) for each $j \in \{0, \dots, 2p-1\}$, the curve

$$ (0, r_1) \ni r \mapsto a_j(r) = (r \cos(\tilde{\omega}_j(r), r \sin(\tilde{\omega}_j(r))) $$

is smooth and has semi-tangent $\omega_j$ at the origin.

**Proof.** To prove (i), we observe that in each interval $\{r\} \times [\omega_j + \alpha_p, \omega_{j+1} - \alpha_p]$, $|W(r, \omega)| \ge \frac{1}{2} \sin(\alpha_1)$. To prove (ii), we observe that the function $W(r, \omega)$ changes sign in $\{r\} \times (\omega_j - \alpha_p, \omega_j + \alpha_p)$ and that its partial derivative with respect to $\omega$ does not vanish. Assertion (iii) follows from the implicit function theorem. Assertion (iv) follows from the previous ones. $\square$

**Remark B.3.** Later on, we will encounter the following situation. There exist two eigenfunctions $v_1$ and $v_2$ such that the functions $h_i = v_i \circ \exp_x$ satisfy the relations

$$ (B.15) \qquad \left\{ \begin{array}{l} h_1(r \cos \omega, r \sin \omega) = r^p \sin(p\omega) + R_{1,p+1}(r \cos \omega, r \sin \omega), \\ h_2(r \cos \omega, r \sin \omega) = r^p \cos(p\omega) + R_{2,p+1}(r \cos \omega, r \sin \omega). \end{array} \right. $$

Defining the family of functions $w_\theta = \cos \theta v_1 - \sin \theta v_2$, the associated family of functions $h_\theta = w_\theta \circ \exp_x$, satisfies

$$ (B.16) \qquad \left\{ \begin{array}{l} h_{\theta}(r \cos \omega, r \sin \omega) = r^p \sin(p(\omega - \theta)) + R_{\theta,p+1}(r \cos \omega, r \sin \omega), \quad \text{with} \\ R_{\theta,p+1} = \cos \theta R_{1,p+1} - \sin \theta R_{2,p+1}. \end{array} \right. $$

Then, Proposition B.2 remains valid for the family $h_\theta$, uniformly with respect to the variable $\theta \in [0, 2\pi]$, with $\omega_j$ replaced by $(\omega_j - \theta)$, and the corresponding functions $r \mapsto \tilde{\omega}_j(r, \theta)$ and $r \mapsto a_j(r, \theta)$ are smooth in $(r, \theta)$.

**Remark B.4.** Proposition B.2 tells us that, in a neighborhood of the critical zero $x$ of $u$, the nodal set $\mathcal{Z}(u)$ consists of $p$ smooth semi-arcs emanating from $x$ tangentially to the rays $\omega_j$.

## APPENDIX C. LOCAL STRUCTURE OF THE NODAL SET NEAR A CRITICAL ZERO, ANOTHER APPROACH

In this Appendix, we use a local conformal chart to determine the local structure of the nodal set $\mathcal{Z}(u)$ near an interior critical zero of $u$, instead of normal coordinates as in Appendix B.

**Remark C.1.** Using “boundary isothermal coordinates” as given in [YaZh2021, Section 2], one can also determine the local structure of the nodal set $\mathcal{Z}(u)$ near a boundary singular point of $u$ (assuming Dirichlet, Neumann or Robin boundary condition).
Let $(M, g)$ be a compact Riemannian surface, and let $V: M \to \mathbb{R}$ be a real valued smooth (ie, $C^\infty$) potential. Let $u \neq 0$, a real valued function, satisfying $(-\Delta+V)u = \lambda u$, for some real number $\lambda$. Let $x$ be a given (interior) point of $M$ (in case $M$ has a boundary). Assume that $u(x) = 0$.

Choose some $r_0 > 0$ such that the geodesic disk $D(x, r_0)$ is contained in a conformal coordinate chart centered at $x$: there exists some open neighborhood $U_x$ of $x$, such that $D(x, r_0) \subset U_x \subset M$, and a conformal diffeomorphism $\phi: U_x \to \mathbb{D}$, such that $\phi(x) = 0$ and $\phi^*(d\xi_1^2 + d\xi_2^2) = \sigma(\psi(\xi_1, \xi_2)) g$. Here $\mathbb{D}$ is the unit disk in $\mathbb{R}^2$, $(\xi_1, \xi_2)$ denote the coordinates in $\mathbb{R}^2$, $\psi = \phi^{-1}$, and $\sigma$ is a smooth positive function on $U_x$. Then, we have

$$ (C.1) \qquad - \Delta_0 f = \sigma (\lambda - V \circ \psi) f, $$

where, $f = u \circ \psi$, and $\Delta_0$ is the Laplacian in $\mathbb{R}^2$ with the metric $d\xi_1^2 + d\xi_2^2$. To determine the local structure of the nodal set $\mathcal{Z}(u)$ in a neighborhood of $x \in \text{int}(M)$, is sufficient to determine the structure of the zero set of $f$ in a neighborhood of 0 in $\mathbb{D}$.

The unique continuation theorem [Aron1957, DoFe1990a] implies that $f$ (equiva-
lently $u$) does not vanish at infinite order at 0. If $\text{ord}(u, x) = \text{ord}(f, 0) = p$, Taylor's
formula at 0, gives

$$ (C.2) \qquad \left\{ \begin{array}{l} f(\xi_1, \xi_2) = \sum_{|\alpha|=p} \frac{1}{\alpha!} D^\alpha f(0,0) (\xi_1, \xi_2)^\alpha + R_{p+1}(\xi_1, \xi_2), \quad \text{where} \\ R_p p + 1(\xi_1, \xi_2) = \sum_{|\alpha|=p+1} \frac{p+1}{\alpha!} (\xi_1, \xi_2)^\alpha \int_0^1 (1-t)^p D^\alpha f(t\xi_1, t\xi_2) dt. \end{array} \right. $$

Here, as usual,

$$ (C.3) \qquad \left\{ \begin{array}{l} \alpha = (\alpha_1, \alpha_2), \quad |\alpha| = \alpha_1 + \alpha_2, \quad (\xi_1, \xi_2)^\alpha = \xi_1^{\alpha_1} \xi_2^{\alpha_2}, \text{ and} \\ D^\alpha f = \frac{\partial^{|\alpha|} f}{\partial \xi_1^{\alpha_1} \partial \xi_2^{\alpha_2}}. \end{array} \right. $$

Using (C.1), and identifying the terms with lowest order, we find that the polynomial $P_p(\xi_1, \xi_2) := \sum_{|\alpha|=p} \frac{1}{\alpha!} D^\alpha f(0,0) (\xi_1, \xi_2)^\alpha$ is homogenous of degree $p$, and harmonic (with respect to $\Delta_0$). Writing the harmonicity condition in polar coordinates $(\rho, \omega)$ in $\mathbb{R}^2$, we find that the polynomial $P_p$ has the form

$$ (C.4) \qquad P_p(\rho \cos \omega, \rho \sin \omega) = a \rho^p \sin(p\omega - \omega_0). $$

for some $0 \neq a \in \mathbb{R}$ and some $\omega_0 \in [0, 2\pi]$.

Multiplying the function $f$ by some constant, and rotating the coordinates $(\xi_1, \xi_2)$ in $\mathbb{R}^2$ if necessary, we can assume that $a = 1$ and $\omega_0 = 0$. It follows that $f$ can be written as

$$ (C.5) \qquad f(\rho \cos \omega, \rho \sin \omega) = \rho^p \sin(p\omega) + R_{p+1}(\rho \cos \omega, \rho \sin \omega), $$

and we can write the second term as

$$ (C.6) \qquad \left\{ \begin{array}{l} R_{p+1}(\rho \cos \omega, \rho \sin \omega) = \rho^{p+1} T_{p+1}(\rho, \omega), \quad \text{where } T_{p+1}(\rho, \omega) \text{ is given by} \\ \qquad \sum_{|\alpha|=p+1} \frac{p+1}{\alpha!} (\cos \omega, \sin \omega)^\alpha \int_0^1 (1-t)^p D^\alpha f(t\rho \cos \omega, t\rho \sin \omega) dt. \end{array} \right. $$

Define

$$ (C.7) \qquad W(\rho, \omega) := \sin(p\omega) + \rho T_{p+1}(\rho, \omega). $$
The function $W(0, \omega)$ vanishes precisely for the values

$$ (C.8) \qquad \omega_j := j \frac{\pi}{p}, \quad j \in \{0, \dots, 2p-1\}. $$

Choose $\alpha_1 \in (0, \frac{\pi}{8})$ and define $\alpha_p := \frac{\alpha_1}{p}$. We have the following relations.

$$ (C.9) \qquad \left\{ \begin{array}{l} \sin(p(\omega_j \pm \alpha_p)) = \pm(-1)^j \sin(\alpha_1), \\ |\sin(p\omega)| \ge \sin\alpha_1, \text{ for } \omega \notin \bigcup_{j=0}^{2p-1} (\omega_j - \alpha_p, \omega_j + \alpha_p) \end{array} \right. $$

Define

$$ (C.10) \qquad \left\{ \begin{array}{l} \rho_0 := \min \left\{ \frac{1}{2}, \frac{1}{2} \sin(\alpha_1) \|T_{p+1}\|_{\infty, D(\frac{1}{2})}^{-1} \right\}, \\ \rho_1 := \min \left\{ \frac{1}{2}, \frac{1}{2} \cos(\alpha_1) \|\partial_\omega T_{p+1}\|_{\infty, D(\frac{1}{2})}^{-1} \right\}, \end{array} \right. $$

where $\| \cdot \|_{\infty, D(\frac{1}{2})}$ denotes the $L^\infty$ norm of functions on the disk $D(\frac{1}{2})$ of radius $\frac{1}{2}$.

**Proposition C.2.** For any $\rho \le \min\{\rho_0, \rho_1\}$,

(i) the function $\omega \mapsto W(\rho, \omega)$ does not vanish in

$$ [0, 2\pi] \setminus \bigcup_{j=0}^{2p-1} (\omega_j - \alpha_p, \omega_j + \alpha_p) = \bigcup_{j=0}^{2p-1} [\omega_j + \alpha_p, \omega_{j+1} - \alpha_p] ; $$

(ii) for each $j \in \{0, \dots, 2p-1\}$, the function $\omega \mapsto W(\rho, \omega)$ has exactly one zero $\tilde{\omega}_j(\rho) \in (\omega_j - \alpha_p, \omega + \alpha_p)$;

(iii) for each $j \in \{0, \dots, 2p-1\}$, the function $\rho \mapsto \tilde{\omega}_j(\rho)$ is $C^\infty$ in the interval $(0, \min\{\rho_0, \rho_1\})$ and tends to $\omega_j$ as $\rho$ tends to zero;

(iv) for each $j \in \{0, \dots, 2p-1\}$, the curve

$$ (0, \min\{\rho_0, \rho_1\}) \ni \rho \mapsto a_j(\rho) = (\rho \cos(\tilde{\omega}_j(\rho)), \rho \sin(\tilde{\omega}_j(\rho))) $$

is smooth and has semi-tangent $\omega_j$ at the origin.

*Proof.* To prove (i), we observe that in each interval $\{\rho\} \times [\omega_j + \alpha_p, \omega_{j+1} - \alpha_p]$, $|W(\rho, \omega)| \ge \frac{1}{2}\sin(\alpha_1)$. To prove (ii), we observe that the function $W(\rho, \omega)$ changes sign in $\{\rho\} \times (\omega_j - \alpha_p, \omega_j + \alpha_p)$ and that its partial derivative with respect to $\omega$ does not vanish. Assertion (iii) follows from the implicit function theorem. Assertion (iv) follows from the previous ones. $\square$

**Remark C.3.** Later on, we will encounter the following situation. There exist two eigenfunctions $v_1$ and $v_2$ such that the functions $h_i = v_i \circ \psi$ satisfy the relations

$$ (C.11) \qquad \left\{ \begin{array}{l} h_1(\rho \cos \omega, \rho \sin \omega) = \rho^p \sin(p\omega) + R_{1,p+1}(\rho \cos \omega, \rho \sin \omega), \\ h_2(\rho \cos \omega, \rho \sin \omega) = \rho^p \cos(p\omega) + R_{2,p+1}(\rho \cos \omega, \rho \sin \omega). \end{array} \right. $$

Defining the family of functions $w_\theta = \cos\theta v_1 - \sin\theta v_2$, the associated family of functions $h_\theta = w_\theta \circ \psi$, satisfies

$$ (C.12) \qquad \left\{ \begin{array}{l} h_{\theta}(\rho \cos \omega, \rho \sin \omega) = \rho^p \sin(p(\omega - \theta)) + R_{\theta,p+1}(\rho \cos \omega, \rho \sin \omega), \text{ with} \\ R_{\theta,p+1} = \cos \theta R_{1,p+1} - \sin \theta R_{2,p+1}. \end{array} \right. $$

Then, Proposition C.2 remains valid for the family $h_\theta$, uniformly with respect to the variable $\theta \in [0, 2\pi]$, with $\omega_j$ replaced by $(\omega_j - \theta)$, and the corresponding functions $\rho \mapsto \tilde{\omega}_j(\rho, \theta)$ and $\rho \mapsto a_j(\rho, \theta)$ are smooth in $(\rho, \theta)$.
**Remark C.4.** Proposition C.2 tells us that, in a neighborhood of the critical zero $x$ of $u$, the nodal set $\mathcal{Z}(u)$ consists of $p$ smooth semi-arcs emanating from $x$ tangentially to the rays $\omega_j$. An alternative proof is to take exponential coordinates at $x$, and polar coordinates in $T_x M$, as in Appendix B. Cheng [Chen1976] gives a more precise, yet less quantitative, result by showing that there exists a local diffeomorphism sending the nodal set of $u$ in a neighborhood of the singular point $x$ onto the nodal set of higher order terms of the Taylor expansion of $u$ at $x$ (a harmonic polynomial) which consists of rays.

APPENDIX D. LABELING LOOPS AND NODAL DOMAINS

In this Appendix, $(M, g)$ is a surface of genus 0 and we consider eigenfunctions of the operator $-\Delta + V$, where $g$ is a $C^\infty$ metric and $V$ a $C^\infty$ real valued potential. Similar arguments can be applied to smooth bounded domains in $\mathbb{R}^2$.

D.1. **Labeling of loops and bouquets of loops.** Let $u$ be an eigenfunction such that $\text{ord}(u, x) = p$, with $x \in \text{int}(M)$. Furthermore, assume that the point $x$ is the sole critical zero of $u$, and that the nodal set $\mathcal{Z}(u)$ is connected.

In a small neighborhood of $x$, the nodal set $\mathcal{Z}(u)$ consists of $2p$ nodal semi-arcs emanating from $x$, tangentially to $2p$ distinct$^7$ rays. Fix an orientated 2-frame in $T_x M$, with first vector supported by one of the rays, denoted by $\omega_0$, and label the $2p$ rays counter-clockwise, $\omega_0, \omega_1, \dots, \omega_{2p-1}$. Denote by $(r, \omega)$ the polar coordinates associated with this frame in $T_x M$.

Since nodal arcs can only meet at critical zeros, the nodal semi-arc emanating from $x$ tangentially to the ray $\omega_j$ must eventually come back to $x$, arriving tangentially to some ray $\omega_{\tau(j)}$, forming a loop $\gamma_{j,\tau(j)}$ at $x$. Hence, we have a map,

$$ (D.1) \qquad \tau : L_0 \mapsto L_0, \text{ where } L_0 := \{0, \dots, 2p-1\}, $$

with the following properties,

$$ (D.2) \qquad \left\{ \begin{array}{l} \tau^2 = \text{Id}, \\ \tau(j) \neq j, \quad \forall j \in L_0, \\ \tau(j) - j \text{ is odd}, \quad \forall j \in L_0. \end{array} \right. $$

The first property is clear. The second and third ones follow from the local structure theorem.

The global picture is that the nodal set $\mathcal{Z}(u)$ is a $p$-bouquet of loops $W(L_0)$ at $x$: $p$ simple loops at $x$ which do not intersect away from $x$, and which meet transversally at $x$.

Take any loop, $\gamma_{j,\tau(j)}$. Exchanging, $j$ and $\tau(j)$ if necessary, we may always assume that $j < \tau(j)$. Consider the sets

$$ (D.3) \qquad \left\{ \begin{array}{l} L_{(j)} := \{j, (j+1), \dots, (\tau(j)-1), \tau(j)\}, \\ J'_{(j)} := L_{(j)} \setminus \{j, \tau(j)\} \end{array} \right. $$

Since $M$ is topologically a sphere, $M \setminus \{\gamma_{j,\tau(j)}\}$ has two components. The local structure of $\mathcal{Z}(u)$ at $x$ shows that the rays $\omega_k, k \in J'_{(j)}$, point inside one of the components, call it $A_{(j)}$. The corresponding nodal arcs must stay in $A_{(j)}$, so that

<footnote>$^7$The rays actually form an equiangular system, but this is not needed here.</footnote>
$J'_{(j)}$ is invariant under $\tau$, and corresponds to a bouquet of loops $W(J'_{(j)}) \subset A_{(j)}$. Similarly, the rays $\omega_k, k \in L_0 \setminus L_{(j)}$, point inside the other component, call it $A'_{(j)}$. The corresponding nodal arcs stay in $A'_{(j)}$, so that $L_0 \setminus L_{(j)}$ is invariant under $\tau$, and corresponds to a bouquet of loops $W(L_0 \setminus L_{(j)}) \subset A'_{(j)}$.

In the polar coordinates $(r, \omega)$, and for $r$ small enough, the nodal semi-arcs emanating from $x$ are given by equations $r \mapsto \exp_x(r\tilde{\omega}_j(r))$, $0 \le j \le 2p-1$. These arcs determine $2p$ intervals

$$ (D.4) \qquad I_j(r) := \{\exp_x(r\omega) \mid \omega \in (\tilde{\omega}_j(r), \tilde{\omega}_{j+1}(r))\}, \quad 0 \le j \le 2p-1, $$

on the circle $S_x(r) = \{\exp_x(r\omega) \mid \omega \in [0, 2\pi]\}$. The function $u$ does not vanish in these open intervals, and changes sign while crossing an end point $\exp_x(r\tilde{\omega}_j(r))$ along the circle. In particular, each open interval is contained in a unique nodal domain, and two contiguous intervals are contained in different nodal domains. Note that the point $x$ is in the closure of each nodal domain of $u$, so that each nodal domain meets at least one interval.

**Lemma D.1.** The complement of a $p$-bouquet of loops $W$ in $M$ has $(p+1)$ components.

*Proof.* Looking at the $p$-bouquet as a connected multi-graph with 1 vertex and $p$ edges, we find that the number of components in the complement of the bouquet $W$ is $\alpha_2(W, \mathbb{S}^2) = \alpha_1(W) - \alpha_0(W) + c(W) + 1 = p+1$ (with the notation of Section 4). Note that a $p$-bouquet of loops is not a graph in the sense of Giblin [Gibl2010]. We can modify the bouquet as in Subsection 4.1 to turn it into a graph, so that we can apply Euler's formula for the sphere. $\square$

**D.2. Labeling of nodal domains.** We now define a labeling of nodal domains.

**Definition D.2.** Label the nodal domains by introducing the map

$$ \delta: L_0 \to \{1, \dots, p\} $$

such that the interval $I_j(r)$ is contained in the nodal domain $\Omega_{\delta(j)}$. More precisely,

(1) let $\Omega_1$ be the nodal domain which contains the interval $I_0(r)$, i.e., $\delta(0) := 1$;

(2) let $\Omega_2$ be the nodal domain which contains the interval $I_1(r)$, i.e., $\delta(1) := 2$;

(3) for $j \ge 1$, assuming that $\delta(1), \dots, \delta(j)$ are defined, let

◇ $\delta(j+1) = \delta(\ell)$, if the interval $I_{j+1}(r)$ is contained in the nodal domain $\Omega_{\delta(\ell)}$ for some $\ell \le j$, i.e. in a nodal domain already labeled;

◇ let $\delta(j+1) = \delta(j) + 1$, if the interval $I_{j+1}(r)$ is not contained in an already labeled nodal domain.

**Property D.3.** Once we have chosen an orientation in $T_x M$, an initial ray $\omega_0$, and labeled the other rays counter-clockwise, the map $\tau: L_0 \to L_0$ determines the map $\delta: L_0 \to \{1, \dots, p\}$, and conversely.

**D.2.1. Working out an example.** Choose $p=8$, so that

$$ L_0 = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}, $$

and define the map $\tau$ in matrix form ($\tau$ maps the first line to the second line) by

$$ (D.5) \qquad \tau = \begin{pmatrix} 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ 3 & 2 & 1 & 0 & 9 & 8 & 7 & 6 & 5 & 4 & 15 & 12 & 11 & 14 & 13 & 10 \end{pmatrix}. $$
Since $\tau(0) = 3$, $\tau(4) = 9$ and $\tau(10) = 15$, we decompose $L_0$ into three subsets
$L_1 = \{0, 1, 2, 3\}$, $L_2 = \{4, 5, 6, 7, 8, 9\}$ and $L_3 = \{10, 11, 12, 13, 14, 15\}$,

$$
(D.6) \quad \tau = \left( \left| \begin{array}{ccccccccccccc} 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ 3 & 2 & 1 & 0 & 9 & 8 & 7 & 6 & 5 & 4 & 15 & 12 & 11 & 14 & 13 & 10 \end{array} \right| \right),
$$

It is clear from the definition of $\tau$ that these subsets are $\tau$-invariant. The reason
behind this fact is explained in Subsection D.1.

◇ Since $\tau(0) = 3$, the first subset we consider is $L_1 = \{0,1,2,3\}$. Let $\ell_1 = 0$,
$k_1 := |L_1|/2 = 2$. The set $L_1$ corresponds to a $k_1$-bouquet of loops, $W(L_1) :=$
$\{\gamma_{0,3}, \gamma_{1,2}\}$. The set $J'_1 := L_1 \setminus \{0,3\} = \{1,2\}$ corresponds to a $(k_1 - 1)$-bouquet of
loops, $W(J'_1) := \{\gamma_{1,2}\}$.

Since $M$ is a sphere, $M\backslash\gamma_{0,3}$ has two components $A_1$ and $A'_1$, and we choose $A_1$ such that $W(J'_1) \subset A_1$ (see Subsection D.1). We begin labeling the nodal domains.

We have $\delta(0) = 1$, $\delta(1) = 2$. Since $k_1 = 2$, $W(L_1)$ contains two loops, $I_0(r)$, $I_1(r)$ and $I_2(r)$ are contained in $A_1$, and we must have $\delta(2) = \delta(0) = 1$. Furthermore, since $A_1$ is simply connected, the set $A_1\setminus W(J'_1)$ has two components which are nodal domains of $u$. It also follows that $I_3(r) \subset A'_1$, which implies that $\delta(3) = k_1 + 1 = 3$. We have actually determined $\delta$ on $L_1$.

◇ Since $\tau(4) = 9$, we turn to the second set, $L_2 = \{4,5,6,7,8,9\}$, with $\ell_2 = 4$, $k_2 = |L_2|/2 = 3$, and $J'_2 = \{5,6,7,8\}$. The set $M\setminus\gamma_{4,9}$ has two components, $A_2$ and $A'_2$, and we choose $A_2$ such that $W(J'_2) \subset A_2$. Notice that the nodal domain $\Omega_3$ (corresponding to $\delta(3)$ is contained in $A'_1 \cap A'_2$. Since $A_2$ is simply connected, the set $A_2\setminus W(J'_2)$ has $k_2 = 3$ components which are nodal domains of $u$. These nodal domains are different from the previous labeled ones ($\Omega_1, \Omega_2$ and $\Omega_3$). We label them beginning with $k_1+2=4$; furthermore $\delta(\ell_2) = \delta(\tau(\ell_2)-1) = k_1+2=4$, i.e. $\delta(4) = \delta(8) = 4$. To continue, we look at $J'_2 = \{5,6,7,8\}$ which is invariant by $\tau$. The action of $\tau$ on this set is similar to the action of $\tau$ on $L_1$. Taking into account the fact that we must have $\delta(5) = k_1+3=5$, we conclude that $\delta(5) = 5, \delta(6) = 6$ and $\delta(7) = 5$. Notice that $\delta(9) = k_1+1=3$. We have labeled $1+k_1+k_2$ nodal domains.

◇ Since $\tau(10) = 15$, we turn to the third set, $L_3 = \{10, 11, 12, 13, 14, 15\}$, with $\ell_3 = 10$, $k_3 = |L_3|/2 = 3$ and $J'_3 = \{11, 12, 13, 14\}$. Notice that the actions of $\tau$ on $L_2$ and $L_3$ are different. The set $M\setminus\gamma_{10,15}$ has two components, $A_3$ and $A'_3$, and we choose $A_3$ such that $W(J'_3) \subset A_3$. Notice that the nodal domain $\Omega_3$ (corresponding to $\delta(3)$ is contained in $A'_1 \cap A'_2 \cap A'_3$. Since $A_3$ is simply connected, the set $A_3\setminus W(J'_3)$ has $k_3 = 3$ components which are nodal domains of $u$. These nodal domains are different from the previous labeled ones $(\Omega_1, \dots, \Omega_6)$. We label them beginning with $k_2+k_1+2=7$; furthermore $\delta(\ell_3) = \delta(\tau(\ell_3)-1) = k_2+k_1+2=7$, i.e. $\delta(10) = \delta(14) = 7$. To continue, we look at the sets $\{11, 12\}$ and $\{13, 14\}$, and we easily infer that $\delta(11) = 8, \delta(12) = 7$, and $\delta(13) = 9$.

We have proved, for this explicit example, that the map $\tau$ determines the map $\delta$.

We can describe the map $\delta$ by the following matrix

$$
(D.7) \quad \delta = \begin{pmatrix} 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ 1 & 2 & 1 & 3 & 4 & 5 & 6 & 5 & 4 & 3 & 7 & 8 & 7 & 9 & 7 & 3 \end{pmatrix},
$$
where $\delta$ sends the entry $j \in L_0$ in the first line, the label of the interval $I_j(r)$, to the entry $\delta(j)$ in the second line, the label of the nodal domain $\Omega_{\delta(j)}$.

To determine $\tau(0)$, we look at the occurrences of 1, the label of the first nodal domain $\Omega_1$: we find $1 = \delta(0) = \delta(2)$, and notice that $\delta(3) = \delta(15) = 3$. We have determined that $L_1 = \{0, 1, 2, 3\}$. This also tells us that the nodal domain $\Omega_2$ is bounded by a single loop. It follows that $\tau$ on $L_1$ is given by

$$ \tau|_{L_1} = \begin{pmatrix} 0 & 1 & 2 & 3 \\ 3 & 2 & 1 & 0 \end{pmatrix}. $$

The next nodal domain which appears is $\Omega_4$. To determine $\tau(4)$, we look at the occurrences of 4, $4 = \delta(4) = \delta(8)$, and we note that $\delta(3) = \delta(9) = 3$. This means that $\tau(4) = 9$, so that $L_2 = \{4, 5, 6, 7, 8, 9\}$. In $A_2$, the label 4 occurs twice, the label 5 occurs twice as well, and the label 6 once. We conclude that

$$ \tau|_{L_2} = \begin{pmatrix} 4 & 5 & 6 & 7 & 8 & 9 \\ 9 & 8 & 7 & 6 & 5 & 4 \end{pmatrix}. $$

The next nodal domain to appear is $\Omega_7$. To determine $\tau(10)$, we look at the occurrences of 7: $7 = \delta(10) = \delta(12) = \delta(14)$, while $3 = \delta(9) = \delta(15)$. This determines $L_3$, and actually $\tau|_{L_3}$ as well,

$$ \tau|_{L_3} = \begin{pmatrix} 10 & 11 & 12 & 13 & 14 & 15 \\ 15 & 12 & 11 & 14 & 13 & 10 \end{pmatrix}. $$

We have recovered the map $\tau$ from the map $\delta$ in the example at hand.

### D.2.2. *Proof of Property D.3.*

* We first show that the map $\tau$ determines the map $\delta$.

◇ We first consider the loop $\gamma_{0,\tau(0)}$. Note that $0 < \tau(0) \le 2p - 3$. Define

$$ (D.8) \qquad \left\{ \begin{array}{l} \ell_1 = 0, \\ L_1 = \{0, \dots, \tau(0)\} \text{ and } L'_1 = L_1 \setminus \{0, \tau(0)\}, \\ k_1 = \frac{1}{2}|L_1| = \frac{1}{2}(\tau(0) + 1). \end{array} \right. $$

The loop $\gamma_{0,\tau(0)}$ divides $M$ into two components, both homeomorphic to a disk (recall that $M$ is topologically a sphere). The rays with index in $J'_1$ point inside one of the components, call this first component $A_1$; the rays with index in $L_0 \setminus L_1$ point inside the second component $A'_1$. In case $J'_1 = \emptyset$, there is no ray pointing into the component $A_1$, and $A_1$ is actually a nodal domain bounded by the loop $\gamma_{0,1}$. Because the loops at $x$ do not intersect away from $x$, the loops with semi-tangents in $J'_1$ must stay inside $A_1$; the loops with semi-tangents in $L_0 \setminus L_1$ must stay inside $A'_1$.

In particular, this implies that the map $\tau$ leaves the sets $L_1$, $J'_1$ and $L_0 \setminus L_1$ globally invariant. To the set $L_1$ we can therefore associate a $k_1$-bouquet $W(L_1)$ of loops at $x$, whose complement in $M$ has $(k_1 + 1)$ components. Similarly, to the set $J'_1$ we can associate a $(k_1 - 1)$-bouquet $W(J'_1)$ which is contained in $A_1$, and whose complement in $A_1$ has $k_1$ components which are actually nodal domains of $u$. The remaining nodal domains are all contained in $A'_1$. The $k_1$ nodal domains contained in $A_1$ are labeled from 1 to $k_1$; the labeling of the nodal domains contained in $A'_1$
begins with $k_1 + 1$. The intervals $I_j(r), 0 \le j \le \tau(0) - 1$ are contained in $A_1$; the intervals $I_j(r), \tau(0) \le j$ are contained in $A'_1$. From these facts, we infer that

$$ (D.9) \qquad \left\{ \begin{array}{l} \delta(0) = \delta(\tau(0) - 1) = 1, \\ \delta(1) = 2, \\ \delta(\tau(0)) = \delta(2p - 1) = k_1 + 1, \\ \delta(\tau(0) + 1) = k_1 + 2. \end{array} \right. $$

◇ Since the labeling of the nodal domains contained in $A_1$ and the labeling of the nodal domains contained in $A'_1$ are essentially independent matters, we can pursue labeling the nodal domains in $A_1$ as if we only worked with the bouquet $W(L_0 \setminus L_1)$.

The domain $A_1$ is simply connected, with boundary $\gamma_{0,\tau(0)}$. Any loop $\gamma$ in the bouquet $W(J'_1)$ is contained in $A_1$ and hence separates $A_1$ into two components, a component $A_\gamma$ which only touches $\gamma_{0,\tau(0)}$ at the point $x$, and a component $A'_\gamma$ one of whose boundaries is $\gamma_{0,\tau(0)}$, and hence meets the nodal set $\Omega_1$.

We now look at the bouquets of nodal loops contained in $A_1$. If $\tau(1) = 2$, the corresponding loop $\gamma_{1,2}$ bounds the nodal domain $\Omega_2$, we have $\delta(3) = 3$, and we are left with analyzing the smaller bouquet associated with the set $\{3, \dots, (\tau(0) - 1)\}$. If $\tau(1) > 2$, we apply the above reasoning with the bouquet associated with the set $\{1, \dots, \tau(1)\}$.

◇ We now define

$$ (D.10) \qquad \left\{ \begin{array}{l} \ell_2 = \tau(\ell_1) + 1 = \tau(0) + 1, \\ L_2 = \{\ell_2, \dots, \tau(\ell_2)\} \text{ and } L'_2 = L_2 \setminus \{\ell_2, \tau(\ell_2)\}, \\ k_2 = \frac{1}{2}|L_2| = \frac{1}{2}(\tau(\ell_2) - \ell_2 + 1). \end{array} \right. $$

We reason as above. The loop $\gamma_{\ell_2,\tau(\ell_2)}$ divides $M$ into two components. One of them $A_2$ contains the $(k_1 - 1)$-bouquet of loops $W(J'_2)$; the other one, denoted by $A'_2$, contains the nodal domains $\Omega_{k_0+1}$. The complement of $W(J'_2)$ in $A_2$ has $k_2$ components which are nodal domains, which are labeled $\Omega_{k_0+2}, \dots, \Omega_{k_0+k_1+1}$. In particular, we have $\delta(k_0 + 1) = k_0 + 2$ and $\delta(k_0 + k_1 + 1) = k_0 + 1$.

We then continue as previously by analyzing smaller and smaller bouquets of loops inside $A_2$.

After finitely many such steps, we exhaust the set $L_0$. This proves that the map $\tau$ determines the map $\delta$.

• We now show that the map $\delta$ determines the map $\tau$.

It is not clear which maps $\delta$ correspond to maps $\tau$ (with the above properties). Assume that we are given some $\delta$ associated with some $p$-bouquet of loops as above. The $p$-bouquet actually defines some map $\tau$ which induces $\delta$. In order to recover $\tau$ from $\delta$, the idea is to recover the sets $L_i$ introduced above, and to reason by induction on the length of the size of the bouquets.

Assume that we are given a map $\delta: L_0 := \{0, \dots, 2p - 1\} \to \{1, \dots, p\}$, induced by some map $\tau: L_0 \to L_0$ such that $\tau^2 = \text{Id}$ and, for all $j \in L_0$, $\tau(j) \ne j$, and $\tau(j) - j$ is odd. Does $\delta$ determine $\tau$?
◇ We first look at the nodal domain $\Omega_1$, with label 1, and at the set $\delta^{-1}(1)$. If $\delta^{-1}(1) = \{0\}$, then we must have $\tau(0) = 1$, and the loop $\gamma_{0,1}$ bounds $\Omega_1$. If $|\delta^{-1}(1)| > 1$, we look at $m_1 := \max \delta^{-1}(1)$. Then, the only possibility is that $\tau(0) = m_1 + 1$. Because of our framework, see Subsection 7.6, the subsets $L_1 := \{0, \dots, (m_1 + 1)\}$, $J_1' := \{1, \dots, m_1\}$, and $L_0 \setminus L_1$ must be $\tau$ invariant. Defining $A_1$ and $A_1'$ as above, we see that there are exactly $k_1$ nodal domains inside $A_1$, where $2k_1 = |L_1|$, and we have $\delta(m_1 + 1) = k_1 + 1$. We can then use induction.
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(pierrehberard@gmail.com) PIERRE BÉRARD, INSTITUT FOURIER, UNIVERSITÉ GRENOBLE ALPES AND CNRS, CS 40700, 38058 GRENOBLE CEDEX 9, FRANCE

(Bernard.Helffer@univ-nantes.fr) BERNARD HELFFER, LABORATOIRE JEAN LERAY, UNIVERSITÉ DE NANTES AND CNRS, AND LMO (UNIVERSITÉ PARIS-SUD), 44322 NANTES CEDEX, FRANCE