Problem 1 (yau-contest 2024)
Section: Geometry and Topology

Let $n>1$ be a positive integer.

(i) Does there exist a map $f : S^{2n} \to \mathbb{CP}^n$ with $\deg(f) \neq 0$? Construct an example or disprove it.

(ii) Does there exist a map $f : \mathbb{CP}^n \to S^{2n}$ with $\deg(f) \neq 0$? Construct an example or disprove it.

Problem 2 (yau-contest 2024)
Section: Geometry and Topology

Let $\Sigma \subset \mathbb{R}^3$ be an embedded surface in $\mathbb{R}^3$. A surface is called minimal if, for any $p \in \Sigma$, we have $\kappa_1(p) + \kappa_2(p) = 0$, where $\kappa_1(p)$ and $\kappa_2(p)$ are the two principal curvatures at $p$. Prove that if $\Sigma$ is closed, then $\Sigma$ cannot be minimal.

Problem 3 (yau-contest 2024)
Section: Geometry and Topology

Let $M$ be a closed, simply connected 6-dimensional manifold. Suppose $H_2(M) = \mathbb{Z}_2$. Prove that the Euler characteristic $\chi(M) \neq -1$.

Problem 4 (yau-contest 2024)
Section: Geometry and Topology

Let $(M,g)$ be a closed oriented $n$-dimensional Riemannian manifold. Let $p \in M$ and $\mathrm{Ric}_p$ be the Ricci curvature tensor at $p$, $S_p$ be the scalar curvature at $p$ which is defined to be $S_p := \frac{1}{n} \, \operatorname{tr}_g(\mathrm{Ric}_p)$. Prove that the scalar curvature $S(p)$ at $p \in M$ is given by

$$S_p \,=\, \frac{1}{\omega_{n-1}} \int_{S^{n-1}} \mathrm{Ric}_p(V,V)\, dS^{n-1},$$

where $\omega_{n-1}$ is the area of the unit sphere $S^{n-1}$ in $T_pM$, $V \in S^{n-1}$ are unit vector fields, and $dS^{n-1}$ is the area element on $S^{n-1}$.

Problem 5 (yau-contest 2024)
Section: Geometry and Topology

Let $S^n$ be the $n$-dimensional sphere with $n \ge 2$, and let $G$ be a finite group that acts freely on $S^n$. Suppose $G$ is non-trivial. Then,

(i) Compute the homotopy groups of the quotient space $\pi_i(S^n/G)$ for $0 \le i \le n$.

(ii) Suppose $n$ is even. Prove that $G$ is isomorphic to $\mathbb{Z}_2$.

(iii) Suppose $n$ is odd. Show that $G$ cannot be isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$ for a prime number.

Problem 6 (yau-contest 2024)
Section: Geometry and Topology

Let $M$ be a closed oriented Riemannian manifold, where $g_t$ is a family of smooth Riemannian metrics smoothly depending on $t \in (-\epsilon, \epsilon)$. Suppose there exists a family of eigenfunctions $f_t$ and eigenvalues $\lambda_t$ smoothly depending on $t$ such that

$$\Delta_{g_t} f_t = \lambda_t f_t,$$

where $\Delta_{g_t}$ is the Laplace-Beltrami operator defined using the Riemannian metric $g_t$. Additionally, assume that $f_0$ is not a constant function. We define $\dot{\lambda} := \left.\frac{d}{dt}\right|_{t=0} \lambda_t$ and $\dot{\Delta} := \left.\frac{d}{dt}\right|_{t=0} \Delta_{g_t}$. Prove the following:

(i) As $\lambda_0$ is an eigenvalue of $\Delta_{g_0}$, let $V_{\lambda_0} := \ker(\Delta_{g_0} - \lambda_0)$ be the eigenspace of $\lambda_0$.

(ii) Let $\Pi : L^2(M, g_0) \to V_{\lambda_0}$ be the orthogonal projection onto the eigenspace. Prove that $\dot{\lambda}$ is an eigenvalue of the operator $\Pi \circ \dot{\Delta} : V_{\lambda_0} \to V_{\lambda_0}$.

(iii) Let $\varphi_t : M \to M$ be a 1-parameter family of diffeomorphisms of $M$ and assume $g_t = \varphi_t^* g_0$. Prove that $\dot{\lambda} = 0$.