数理解析研究所講究録
第 1896 巻 2014 年 79-97

79

On superquadratic Dirac equations
on compact spin manifolds
東京工業大学大学院理工学研究科数学専攻
磯部健志

Takeshi Isobe
Department of Mathematics
Graduate School of Science and Engineering
Tokyo Institute of Technology

1Introduction

In this note, we report our recent work about Morse theory for superquadratic
Dirac equations on compact spin manifolds. The details will appear in [18].
Let $(M, g, \rho)$ be an $m$ -dimensional compact Spin manifold, where is a Riemannian metric on $M,$ : $P_{Spin}(M)arrow P_{SO}(M)$ is a spin structure on $M$ . We denote by
the spinor bundle. It is a vector bundle associated to$P_{Spin}(M)arrow M$ via the fundamental spin representation: Spin $(m)arrow$ Aut.
The Dirac operator $D_{g}:C^{\infty}(M, \mathbb{S}(M))arrow C^{\infty}(M, \mathbb{S}(M))$ is define by$g$$\rho$$\mathbb{S}(M)=P_{Spin}(M)\cross_{\sigma}\mathbb{S}_{m}arrow M$$(\mathbb{S}_{m})$$\sigma$$D_{g}:=c\circ\nabla$:$C^{\infty}(M, \mathbb{S}(M))arrow C^{\infty}(M, T^{*}M\nabla\otimes \mathbb{S}(M))$,$\cong C^{\infty}(M, TM\otimes \mathbb{S}(M))arrow cC^{\infty}(M, \mathbb{S}(M))$

is the canonical lift of the Levi-Civita connection on $P_{SO}(M)$ via the double
covering $P_{Spin}(M)arrow P_{SO}(M)$ and is the Clifford multiplication.
We consider nonlinear Dirac equations of the following form:

where$\nabla$$c$$D_{g}\psi=h(x, \psi)$

on $M$ ,(1.1)

where : $S(M)arrow \mathbb{S}(M)$ is a fiber preserving map of the form $h(x, \psi)=\nabla_{\psi}H(x, \psi)$ ,the vertical gradient of $H$ (the dual of) and $H=$
w.r.t the metric on$H(x, \psi)$ a smooth function on.
Eq (1.1) has a variational structure:
is a solution to (1.1) if and only if is a
critical point ofdefined by$h$$d_{\psi}H$$\mathbb{S}(M)$$\mathbb{S}(M)$$\psi$$\psi$$\mathcal{L}_{H}$$\mathcal{L}_{H}(\psi)=\frac{1}{2}\int_{M}\langle\psi, D_{g}\psi\rangle dvo1_{g}-\frac{1}{p+1}\int_{M}H(x, \psi)dvo1_{g}.$We give in the following two examples which partially motivate our study of equations of the form (1.1).

80
Example 1: Dirac harmonic maps (Spinorial version of SUSY -model)
be a Riemann surface andThis model was first introduced by [10], [9]. Let$(N, h)$ a Riemannian manifold. In this model, we consider a pair of two fields,. In components, we write
and
$N$
.
where
a local coordinate system on and
The action functional for supersymmetric Dirac-harmonic maps is defined by$\sigma$$(\Sigma, g)$$\phi\in$$\psi\in C^{\infty}(\Sigma, \mathbb{S}(\Sigma)\otimes\phi^{*}TN)$$C^{\infty}(\Sigma, N)$$\psi=\psi^{k}\otimes\frac{\partial}{\partial y}\tau(\phi)$$\psi_{k}\in C^{\infty}(\mathbb{S}(\Sigma))$$y^{k}$$\mathcal{L}(\phi, \psi)=\frac{1}{2}\int_{\Sigma}|d\phi|^{2}dvo1_{g}+\frac{1}{2}\int_{\Sigma}\langle\psi, D_{\phi}\psi\rangle dvo1_{g}$$- \frac{1}{12}\int_{\Sigma}R_{\dot{n}kjl}(\phi)\langle\psi^{i}, \psi^{j}\rangle\langle\psi^{k}, \psi^{l}\rangle dvo1_{g},$is the Dirac operator associated to the natural connection
the curvature tensor of $(N, h)$ .
and
The Euler-Lagrange equation for the action takes the following form:

where$\nabla^{\phi}$$D_{\phi}=c\circ\nabla^{\phi}$$\mathbb{S}(\Sigma)\otimes\phi^{*}TN$on$R_{\tau jkl}$$\tau^{m}(\phi)-\frac{1}{2}R_{lij}^{m}(\phi)\langle\psi^{i},$$\nabla\phi^{l}\cdot\psi^{j}\rangle+\frac{1}{12}g^{mp}R_{ikjl;p}(\phi)\langle\psi^{i},$ $\psi^{j}\rangle\langle\psi^{k},$ $\psi^{l}\rangle=0$$D_{\phi}\psi^{m}=\frac{1}{3}R_{jkl}^{m}(\phi)\langle\psi^{j}, \psi^{l}\rangle\psi^{k}$

,

,

(1.2)
(1.3)

where
The main characteristic of this problem is the following:is conformally invariant.
As a result, (1.2), (1.3) are conformally invariant equations and depend only on the.
confomal structure of$\tau(\phi)=tr\nabla d\phi.$$\bullet$$\mathcal{L}$$(\Sigma, g)$is quartic in
Combined with the fact thathave:$\bullet \mathcal{L}$.

$\psi.$$H^{1/2}(\Sigma)\subset L^{4}(\Sigma)$

The variational problem associated to$\mathcal{L}$is continuous, but not compact, we

is non-compact and strongly indefinite.

At present, there are no general existence results for (1.2), (1.3) from a variational
point of view. Note, however, that there is a variational theory for the 1-dimensionalcase, the so-called Dirac-geodesics, see [16].
Example 2: Spinorial Yamabe type equations
Let $(M, g, \rho)$ be a compact spin manifold. We assume $H\in C^{\infty}(M)$ is given. We
consider the following action functional:$\mathcal{L}(\psi)=\frac{1}{2}\int_{M}\langle\psi, D_{g}\psi\rangle dvo1_{g}-\frac{m-1}{2m}\int_{M^{H(x)|\psi|^{\frac{2m}{m-1}}}}d_{V}\circ 1_{g}.$The Euler-Lagrange equation of this action is$D_{g}\psi=H(x)|\psi|^{\frac{2}{m-1}}\psi$

.

(1.4)

The equation (1.4) is related to the existence of conformal immersion
with mean curvature $H$ . The following are main characteristic of this problem:$Marrow \mathbb{R}^{m+1}$

81
is conformally invariant.
Thus, (1.4) is a conformally invariant equation and depends only on the conformal
structure of $(M, g)$ .
is the critical power.
$H^{1/2}(M)\subset L^{\frac{2m}{m-1}}(M)$is,Thatis continuous, but not compact. As a result, theassociated variational problem is critical and strongly indefinite.
$\backslash \mathcal{L}$$|\psi|^{\frac{2m}{m-1}}$

Partial existence results were previously established by [21] and [17]. However, theproblem remains widely open in general.

Both of examples 1,2 are non-compact, strongly indefinite variational problems. We want to establish a general variational framework to treat such problems.
Some compactness issues were treated in [15]. In the work [18], we focus on theindefinite variational character of the problem. In this direction, we are especiallyinterested in two topics:(1) Relative Morse indices and its connection with compactness property.
(2) Construction and computation of Morse-Floer homology of $H^{1/2}(M, \mathbb{S}(M))$ associated to$\mathcal{L}_{H}.$2$A$

superquadratic subcritical problem

We first introduce functional setting of the problem. natural function space is$:=H^{1/2}(M, \mathbb{S}(M))$ . It is defined as:-spinors on $M$ denoted by$A$$H^{1/2}$$\mathcal{H}^{1/2}(M)$$\psi\in \mathcal{H}^{1/2}(M) \Leftrightarrow \psi\in L^{2}(M), |D_{g}|^{1/2}\psi\in L^{2}(M)$$\mathcal{H}^{1/2}(M)$.

is a Hilbert space with the following inner product$(\psi, \varphi)_{H^{1/2}}:=(|D_{g}|^{1/2}\psi, |D_{g}|^{1/2}\varphi)_{L^{2}}+(\psi, \varphi)_{L^{2}}.$

We have the Sobolev embedding:$H^{1/2}(M)\subset L^{p+1}(M)$for $0 \leq p\leq\frac{m+1}{m-1}.$

The embedding is compact for, but not for
called the critical exponent.
For $H$ , we assume the following condition:$0 \leq p<\frac{m+1}{m-1}$$\mathcal{L}_{H}$$p+1= \frac{2m}{m-1}$

is

(2.1)

$|H(x, \psi)|\leq C(1+|\psi|^{p+1})$

for some $1<p< \frac{m+1}{m-1}$ . Under the condition (2.1),
We further assume $H(x, \psi)$ is
and satisfies$p= \frac{m+1}{m-1}.$

is a subcritical functional.

$C^{2}$$|d_{\psi\psi}^{2}H(x, \psi)|\leq C(1+|\psi|^{p-1})$

Under the condition (2.2),$\mathcal{L}_{H}$is$C^{2}$

on$\mathcal{H}^{1/2}(M)$

.

(2.2)

. Note that (2.1) follows from (2.2).

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We also assume the following superquadratic condition:

.

(2.3)

$\mathcal{H}^{1/2}(M)$for the class of $H$$2H(x, \psi)+C_{1}|\psi|^{p+1}-C_{2}\leq\langle\psi, H_{\psi}(x, \psi)\rangle$

A model example:$H(x, \psi)=\frac{1}{p+1}H(x)|\psi|^{p+1},$$H(x)>0,$ $H\in C^{0}(M)$ .

We want to estabhsh a Morse theory forsatisfying (2.2) and (2.3).

3$\mathcal{L}_{H}$on

Relative Morse indices and compactness

To do Morse theory, we first need to define Morse index for a critical point of theis
fUnctional. Classically, Morse index (co-index) at a critical point
on whichdefined as the dimension of the maximal subspace of$\psi\in \mathcal{H}^{1/2}(M)$$d^{2}\mathcal{L}_{H}(\psi)<$$\mathcal{H}^{1/2}(M)$$0(>0)$ , where$d^{2} \mathcal{L}_{H}(\psi)(\varphi, \varphi)=\int_{M}\langle\varphi, D_{g}\varphi\rangle dvo1_{g}-\int_{M}\langle H_{\psi\psi}(x, \psi)\varphi, \varphi\rangle dvo1_{g}.$.
Equivalently, it is the dimension of the negative eigenspaces ofNote that $Spec(D_{g})$ is unbounded from below and above. This implies that theat any critical point. Thus the classical Morse
Morse index and co-index are. We need a renormalized version ofontheory does not make sense for$D_{g}-H_{\psi\psi}(x, \psi)$$+\infty$$\mathcal{L}_{H}$$\mathcal{H}^{1/2}(M)$the classical Morse theory.

3.1

Relative Morse indices

To construct right Morse theory, we need renormalized Morse indices. In thefollowing, we introduce three well-known definitions of such renormalized Morse
indices.

1. Relative Morse index as relative dimension
with some
The idea of this definition is to compare the negative space ofbe subspaces. Following [1], [2], we say $W$
fixed subspace. Let $V,$$d^{2}\mathcal{L}_{H}(\psi)$$V,$$W\subset \mathcal{H}^{1/2}(M)$is the orthogonal:commensurable if $P_{V}-P_{W}$ is compact, whereis defined similarly. For such commensurable subspaces
projection onto V.
$W$ , we define the relative dimension $\dim(V, W)$ as$P_{V}$$\mathcal{H}^{1/2}(M)arrow V$$P_{W}$$V,$$\dim(V, W)=\dim(V\cap W^{\perp})-\dim(V^{\perp}\cap W)$

Note that the$H^{1/2}$-self-adjoint realization of$1)^{-1}D_{g}-(|D_{g}|+1)^{-1}H_{\psi\psi}(x, \psi)$.

$d^{2}\mathcal{L}_{H}(\psi)$.

is given by$d^{2}\mathcal{L}_{H}(\psi)=(|D_{g}|+$83
We define, the negative eigenspace ofdefineand.
With these definitions, we give the following$E_{H}^{-}(\psi)=E^{-}(d^{2}\mathcal{L}_{H}(\psi))$$D_{\lambda}=D_{g}-\lambda(\lambda\in \mathbb{R})$Definition 1$\lambda$. We also$E_{\lambda}^{-}=E^{-}((|D_{g}|+1)^{-1}D_{\lambda})$-relative Morse index of$\mathcal{L}_{H}$at$\psi\in \mathcal{H}^{1/2}(M)$$m_{\lambda}(\psi) :=\dim(E_{H}^{-}(\psi), E_{\lambda}^{-})$

Note that sincewell-defined, i.e.,$d^{2}\mathcal{L}_{H}(\psi)$$d^{2}\mathcal{L}_{H}(\psi)-(|D_{g}|+1)^{-1}D_{\lambda}$

is defined as.

is compact, the above definition is$m_{\lambda}(\psi)\in \mathbb{Z}.$2. Relative Morse index as spectral flowBy$d^{2}\mathcal{L}_{H}(\psi)=D_{g}-H_{\psi\psi}(x, \psi):L^{2}(M,\mathbb{S}(M))arrow$is the -self-adjoint realization of.
$:=H_{\psi\psi}(x, \psi),$
We set. It is a symmetric endmorphism of. We define$:=D_{g}-A$ for $A\in \mathcal{A}=L^{\infty}(M, Sym(S(M)))$ .
Let us consider a continuous pathconnectingand. It is afact that the eigenvalues of a generic path
is simple. The spectral flow
sf
is defined as$d^{2}\mathcal{L}_{H}(\psi)(\varphi, \varphi)=((D_{g}-H_{\psi\psi}(x, \psi))\varphi, \varphi)_{L^{2}},$$L^{2}(M, \mathbb{S}(M))$$L^{2}$$\mathbb{S}(M)$$d^{2}\mathcal{L}_{H}(\psi)$$\mathbb{S}(M)arrow \mathbb{S}(M)$$A_{\psi}$$D_{A}$$\{D_{A_{t}}\}_{t\in[0,1]}$$D_{\lambda}$$D_{A_{\psi}}$$\{D_{A_{t}}\}_{t\in[0,1]}$$\{D_{A_{t}}\}_{t\in[0,1]}$$sf\{D_{A_{t}}\}_{t\in[0,1]}$$=$

the number of eigenvalues flowing from negative to positive-the number of eigenvalues flowing from positive to negative.

With these, we give
Definition 2 Relative Morse index$\mu_{\lambda}(\psi)$is defined as$\mu_{\lambda}(\psi)=-sf\{D_{A_{t}}\}_{t\in[0,1]}.$

3. Relative Morse index as Fredholm index
We consider the negative gradient flow connecting two critical points x, y
:

$\in$

crit

$(\mathcal{L}_{H})$$:=$$\{\psi_{\in \mathcal{H}^{1/2}}(M):d\mathcal{L}_{H}(\psi)=0\}$$\frac{\partial\psi}{\partial t}=-\nabla_{1/2}\mathcal{L}_{H}(\psi)$,$\psi(-\infty)=x,$ $\psi(+\infty)=y$

,

(3.1)

where $\nabla_{1/2}\mathcal{L}_{H}(\psi)=(|D_{g}|+1)^{-1}D_{g}-(|D_{g}|+1)^{-1}\nabla_{\psi}H(x, \psi)$ is the-gradientof
It is a fact that (3.1) is Fredholm if x, y are non-degenerate, see [2]. The Fredholm
index of (3.1) is the Fredholm index of the linearization:$H^{1/2}$$\mathcal{L}_{H}.$$\frac{\partial u}{\partial t}=-d\nabla_{1/2}\mathcal{L}_{H}(\psi)u, u(-\infty)=0, u(+\infty)=0.$

Theorem 1 The Fredholm index of (3.1) depends only on
We denote it by $\mu(x, y)$ .

$d\nabla_{1/2}\mathcal{L}_{H}(x)$and$d\nabla_{1/2}\mathcal{L}_{H}(y)$.

84
For the proof of the above theorem, see [6], [4]. The three indicesand $\mu(x, y)$ are related as follows:
Theorem 2 Assume
following hold:(1)$\psi\in crit(\mathcal{L}_{H})$is non-degenerate and$m_{\lambda}(\psi),$$\mu_{\lambda}(\psi)$$\lambda\in \mathbb{R}\backslash Spec(D_{g})$. The$m_{\lambda}(\psi)=\mu_{\lambda}(\psi)$(2) $\mu(x,y)=m_{\lambda}(x)-m_{\lambda}(y)$For the proof of (1), see [19]. For the proof of (2), see [6], [4].

3.2

$A$

compactness theorem via relative Morse indicesWe assume thatgroup$\mathcal{L}_{H}$is Morse onby$\mathcal{H}^{1/2}(M)$and$\mathbb{F}$is a field. We define a graded$\{C_{p}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M))\}_{p\in \mathbb{Z}}$$C_{p}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M))= \oplus \mathbb{F}\langle\psi\rangle,$$\psi\in crit_{p}(\mathcal{L}_{H})$

crit
where
is not necessarily finitely gencompact,
not
is
Since
erated. But, it is the case for some class of $H$ including lower order perturbationsof the model example given in \S 2. Our first result is a compactness result undermore
the relative Morse index bounds. We only state it for the model case.
general result will be found in [18].
$(\mathcal{L}_{H}):m_{\lambda}(\psi)=p\}.$$crit_{p}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M))=\{\psi\in$$C_{p}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M))$$\mathcal{H}^{1/2}(M)$$A$

Theorem 3 Let $m\geq 3$ . Assume that $H(x, \psi)=\frac{1}{p+1}H(x)|\psi|^{p+1}$ , where $H\in C^{0}(M)$with $H>0$ on $M$ and $1<p< \frac{m+1}{m-1}$ . The following assertions are equivalent for ato (1. 1).
sequence of solutions$\{\psi_{n}\}_{n=1}^{\infty}\subset \mathcal{H}^{1/2}(M)$(1)$\sup_{n\geq 1}\mathcal{L}_{H}(\psi_{n})<+\infty.$(2) $\sup_{n\geq 1}m_{\lambda}(\psi_{n})<+\infty.$

Remark 1 (1) In [8], Bahri-Lions considered equations of the $form-\Delta u=a(x)|u|^{p-1}u$under the -Dirichlet boundary condition and obtained similar result,inwhere the Morse index is the classical one.
Angenent-van der Vorst $[7J$ considered equations $-\Delta u=a(x)|v|^{p-1}v,$ $-\Delta v=$$b(x)|u|^{q-1}u$ inunder the -Dirichlet boundary conditions. In this case,these equations arise as a critical point of an indefinite functional. Thus the Morse
index is the relative Morse indices. Thus it is related with our work. In fact, we owe
much to their ideas in our proof of Theorem 3. In this case, however, by the special
structure of the equation, we can eliminate one of the functions, for example , fromthe 2nd equation, and the problem is reduced to an ordinary (classical) index problem.

$\Omega\subset \mathbb{R}^{m}$$0$$\Omega\subset \mathbb{R}^{m}$

$0$

$v$

85
of a single function. This approach can not be applied to our problem. Thus, we willtake another approach.
(2) From Theorem 3, we have thatanyis relatively compact for$crit_{j\mathcal{C}}(\mathcal{L}_{H})\subset \mathcal{H}^{1/2}(M)$$p\in \mathbb{Z}.$3.3

Idea of the Proof of Theorem 3, the first part(1)
is a consequence of $PS$ and continuity of, see [14]. To prove (2)(1), following Bahri-Lions and Angenent-van der Vorst, we argue by contradiction.
Thus, we assume that there exists, a sequence of solutions$\Rightarrow(2)$$\Rightarrow$$m_{\lambda}$$\{\varphi_{n}\}_{n=1}^{\infty}$

on $M$$D_{g}\varphi_{n}=H(x)|\varphi_{n}|^{p-1}\varphi_{n}$(3.2)

such that(3.3)$\mathcal{L}_{H}(\varphi_{n})arrow\infty(\Leftrightarrow\Vert\varphi_{n}\Vert_{L(M)}\inftyarrow+\infty)$

and bounded relative Morse indices$m_{\lambda}(\varphi_{n})\leq k$

We conformal blow-up of : Define$g$$\psi_{n}=\rho_{n}^{-\frac{1}{p-1}}F(\varphi_{n})$a fiberwise isometry.

(3.4)$(M, \rho_{n}^{2}g :=g_{n}),$in$(M, g_{n})_{narrow\infty}arrow(\mathbb{R}^{m}, g_{\mathbb{R}^{m}})$Define.
$\rho_{n}$$:=\Vert\psi_{n}\Vert_{L(M)}^{p-1}\infty$$C_{1oc}^{\infty}(\mathbb{R}^{m})$. We then have.

on $\mathbb{S}(M, g_{n})arrow(M, g_{n})$ , where $F$ : $S(M, g)arrow \mathbb{S}(M, g_{n})$ is

We note the following Conformal property of the Dirac operator:$D_{9n}F(\varphi)=F(\rho_{n}^{-\frac{m+1}{2}}D_{g}(\rho^{\frac{m-1}{n^{2}}}\varphi))=\rho_{n}^{-1}F(D_{g}\varphi)$Thus,$\psi_{n}$.

satisfieson $(M, g_{n})$ ,$D_{g_{n}}\psi_{n}=H(x)|\psi_{n}|^{p-1}\psi_{n}$

(3.5)
(3.6)

$\Vert\psi_{n}\Vert_{L(M,g_{n})}\infty=1$

and$m_{\rho_{n}^{-1}\lambda}(\psi_{n})\leq k$

.

(3.7)

By (3.5), (3.6) we have (after a further renormalization): There exists
such thatin,$\psi_{n}arrow\psi_{\infty}$$($$\psi_{\infty}\in L^{\infty}(\mathbb{R}^{m}, \mathbb{S}(\mathbb{R}^{m}))$$L_{1oc}^{\infty}(\mathbb{R}^{m}))$$D_{g_{\mathbb{R}^{m}}}\psi_{\infty}=H(x_{\infty})|\psi_{\infty}|^{p-1}\psi_{\infty}$$\Vert\psi_{\infty}\Vert_{L\infty(\mathbb{R}^{m})}=1$.

on$\mathbb{R}^{m}$

,

(3.8)
(3.9)

By (3.7), we want to assert$m_{0}(\psi_{\infty})\leq k$.

(3.10)

86
But,$m_{0}(\psi_{\infty})$

has not been defined yet!

Three definitions of the relative Morse indices we have been given before depend
is notbecausecrucially on the compactness of $M$ . So, these do not apply tocompact.
$\bullet$$\mathbb{R}^{m}$$\psi_{\infty}$

To obtain (3.10) from (3.7), we need another formulation of the relative Morsewhich can be applied to non-compact setting as well.
index$m_{\lambda}$

Relative Morse index

3.4$m_{\mathbb{R}^{m}}$and its property

A natural requirement for the “relative Morse index”$m\mathbb{R}^{m}$is the following:-solutions to (3.8) on
is defined for( -1)( -2) It is a natural extension of-convergence (as in the previous( -3) It is (lower-semi)continuous w.r. thesubsection).
$I$$\mathbb{R}^{m}.$$L^{\infty}$$m_{\mathbb{R}^{m}}$$I$$m_{\lambda}.$

$t$

$I$

$L_{1oc}^{\infty}(\mathbb{R}^{m})$We will construct such index in the following. In addition to the above threeproperties, we show that our index has the following property ( -4); in particular,
it is non-trivial.
$I$

Theorem 4 (( -4)) Assume that $m\geq 3$ and $1<p< \frac{m+1}{m-1}$ . For any nontrivial, we haveontosolution$I$$\psi\in L^{\infty}(\mathbb{R}^{m}, \mathbb{S}(\mathbb{R}^{m}))$$m_{\mathbb{R}^{m}}(\psi)=+\infty.$$\mathbb{R}^{m}$$D_{g_{R^{m}}}\psi=|\psi|^{p-1}\psi$

Remark 2 (i) Theorem 4 gives a positive answer to the conjecture by Maalaoui $[20J.$(ii) ( -2) is in fact not necessary. It only motivates our construction of$m\mathbb{R}^{m}.$$I$with the properties $(I-1)-(I-4)$ , Theorem 3
Once we assume the existence ofeasily follows: We have a contradiction from ( -3), ( -4) and (3.10):$m_{\mathbb{R}^{m}}$

$I$

$I$

$+\infty=m_{\mathbb{R}^{m}}(\psi_{\infty})\leq k.$

口

4
4.1

$A$

$A$

construction of the relative Morse index
reformulation of the relative Morse index$m_{\mathbb{R}^{m}}$$m_{\lambda}$

As we have observed, we need a reformulation of the relative Morse indices which) as well. Our reformulation
can be extended to non-compact manifolds (such as
was inspired from the work of J. J. Dusitermaat [12] about classical mechanics.
$\mathbb{R}^{m}$

An example from classical mechanics
We consider classical mechanics on compact manifold $M$ . We denote by $\Lambda(M)=$the free loop space on $M$ . Let $L(t, q, v)$ be a smooth function on $TM,$$C^{\infty}(S^{1}, M)$87where $q\in M$ and $v\in T_{q}M$ . It is called a Lagrangian on $TM$ . We assume thatis convex with respect to . For $q\in\Lambda(M)$ , we define the action functional of
as $\mathcal{L}(q)=\int_{S^{1}}L(t, q(t),\dot{q}(t))dt$. For such , there corresponds to a Hamiltonian
function $H$ on $T^{*}M$ defined by $H(t, q,p)= \max_{v}(\langle q, v\rangle-L(t, q, v))$ . $H$ is theso called the Legendre dual of . For the Hamiltonian $H$ , we define the action$\mathcal{A}(x)=\int_{S^{1}}x^{*}\lambda-H(t, x(x))dt$ defined fora loop on $T^{*}M,$ $x(t)=(q(t),p(t))\in$, whereis the Liouville form.
We have the following correspondence:
$L$

$v$

$L$

$L$

$L$

$x$

$C^{\infty}(S^{1}, T^{*}M)$$\lambda=p_{i}dq^{i}$

is a critical point of$q$$\mathcal{L}rightarrow X1:1=(q,p),$$p=\partial_{v}L(t, q,\dot{q})$is a critical point of$\mathcal{A}.$By J. J. Dusitermaat [12], we also have:

The Morse index of$\mathcal{L}$at $q\in$ crit$(\mathcal{L})=$the Maslov index of $x=(q,p)\in$ crit$(\mathcal{A})$.

We do not mention about what the Maslov index is, but in our case it is thesame as the relative Morse index after a normalization. We expect that the similar
relation holds for our case: We expect$m_{\lambda}(\psi)=$the Morse index of a “dual action$\mathcal{L}^{*}$”

at the dual critical point.

Dual action

Assume that $H(x, \psi)$ is strictly convex innatural candidate for a dual function$H.$
is the Legendre-Fenchel dualof
We consider our model example $H(x, \psi)=\frac{1}{p+1}H(x)|\psi|^{p+1}$ . In this case, however, the dual action will not be. Thus, we can not define the Morse index of
the dual action. To define Morse index, however, the second order information of
the functional is sufficient. Thus, we consider the dual action of the second orderapproximationof. It is defined as$\psi.$$A$$\mathcal{L}_{H}^{*}$$C^{2}$$\mathcal{A}_{\psi,H}$$\mathcal{L}_{H}$$\mathcal{A}_{\psi,H}(\varphi):=\frac{1}{2}d^{2}\mathcal{L}_{H}(\psi)(\varphi, \varphi)$$= \frac{1}{2}\int_{M}\langle\varphi, D_{-\lambda}\varphi\rangle dvo1_{g}-\frac{\lambda}{2}\int_{M}|\varphi|^{2}dvo1_{g}-\frac{1}{2}\int_{M}H(x)|\psi|^{p-1}|\varphi|^{2}dvo1_{g}$$- \frac{p-1}{2}\int_{M}H(x)|\psi|^{p-3}|\langle\psi, \varphi\rangle|^{2}dvo1_{g}$,$= \frac{1}{2}\langle D_{-\lambda}\varphi, \varphi\rangle_{H^{-1/2}\cross H^{1/2}}-G_{H,\lambda}(\varphi)$

where$G_{H,\lambda}$: $L^{2}(M, \mathbb{S}(M))arrow L^{2}(M, \mathbb{S}(M))$ is defined by$G_{H,\lambda}( \varphi)=\frac{\lambda}{2}\int_{M}|\varphi|^{2}dvo1_{g}+\frac{1}{2}\int_{M}H(x)|\psi|^{p-1}|\varphi|^{2}dvo1_{g}$$+ \frac{p-1}{2}\int_{M}H(x)|\psi|^{p-3}|\langle\psi, \varphi\rangle|^{2}dvo1_{g}.$88
The Legendre-Fenchel dual of$G_{H,\lambda}$

is defined by$G_{H,\lambda}^{*}( \varphi):=\max\{\langle\phi, \varphi\rangle_{L^{2}\cross L^{2}}-G_{H,\lambda}(\phi):\phi\in L^{2}(M,\mathbb{S}(M))\}.$

An easy computation shows that
$G_{H,\lambda}^{*}( \varphi)=\frac{1}{2}\int_{M}\frac{1}{\lambda+pH(x)|\psi|p-1}|P_{\psi}(\varphi)|^{2}dvo1_{g}+\frac{1}{2}\int_{M}\frac{1}{\lambda+H(x)|\psi|p-1}|P_{\psi}^{\perp}(\varphi)|^{2}dvo1_{g},$

where
and
The dual action$P_{\psi},$$P_{\psi}^{\perp}\in L^{\infty}(M, Sym(\mathbb{S}(M))$$P_{\psi}^{\perp}=1_{\mathbb{S}(M)}-P_{\psi}$

are defined as$P_{\psi}(x)=$the orthogonal projection on, respectively.
is defined asof$\mathcal{A}_{\psi,H,\lambda}^{*}$$\mathcal{A}_{\psi,H}$$\mathcal{A}_{\psi,H,\lambda}^{*}(\varphi)=G_{H}^{*}(\varphi)-\frac{1}{2}\langle K_{\lambda}\varphi,$$\varphi\rangle_{L^{2}xL^{2}}$$= \frac{1}{2}\int_{M}\frac{1}{\lambda+pH(x)|\psi|p-1}|P_{\psi}(\varphi)|^{2}dvo1_{g}+\frac{1}{2}\int_{M}\frac{1}{\lambda+H(x)|\psi|p-1}|P_{\psi}^{\perp}(\varphi)|^{2}dvol_{g}$$- \frac{1}{2}\int_{M}\langle K_{\lambda}\varphi, \varphi\rangle dvo1_{g},$where
We have:$K_{\lambda}=D_{-\lambda}^{-1}(-\lambda\not\in Spec(D_{g}))$.

Theorem 5 (Index formula I) Assume $\lambda>0,$ $-\lambda\not\in Spec(D_{g})$ . We have$m_{-\lambda}(\psi)=the$

Morse index of$\mathcal{A}_{\psi,H,\lambda}^{*}.$

behaves badly along the blow-up sequence
As we will see shortly,
defined in the previous subsection: The terms $\lambda+pH(x)|\psi|^{p-1}$ and $\lambda+H(x)|\psi|^{p-1}$
are not pleasant for our purposes. In the presence of these, the continuity
inproperty ( -3) is difficult to obtain with the formula of Theorem 5.
is written as
To remedy this, we observe that$\psi=\psi_{n}$$\mathcal{A}_{\psi,H,\lambda}^{*}$$\mathcal{A}_{\psi,H,\lambda}^{*}$$I$$\mathcal{A}_{\psi,H,\lambda}^{*}$$\mathcal{A}_{\psi,H,\lambda}^{*}=((H(x)|\psi|^{p-1}+\lambda L_{\psi})^{-1}L_{\psi}(\varphi), \varphi)_{L^{2}}-(K_{\lambda}(\varphi), \varphi)_{L^{2}},$

where$L_{\psi}= \frac{1}{p}P_{\psi}+P_{\psi}^{\perp}.$

After multiplication by $(H(x)|\psi|^{p-1}+\lambda L_{\psi})^{1/2}$ , we have the following similarity
relation:after multiplication$A_{\psi,H,\lambda}^{*} \cong L_{\psi}-T_{\psi,H,\lambda},$whereat the following:$T_{\psi,H,\lambda}=(H(x)|\psi|^{p-1}+\lambda L_{\psi})^{1/2}\circ K_{\lambda}o(H(x)|\psi|^{p-1}+\lambda L_{\psi})^{1/2}$

Theorem 6 (Index formula II) Assume$\lambda\geq 0,$$-\lambda\not\in Spec(D_{g})$$m_{-\lambda}(\psi)=\neq\{\mu\in Spec(L_{\psi}^{-1}T_{\psi,H,\lambda}):\mu>1\}.$.

. We thus arrive$\langle\psi(x)\rangle$

89
Idea of the proof: For$0\leq\theta\leq 1$, define$A_{\lambda,\psi,\theta}:=\theta[(H(x)|\psi|^{p-1}+\lambda)1+(p-1)H(x)|\psi|^{p-1}P_{\psi}],$We have a correspondence$D_{-\lambda}-A_{\lambda},\psi_{\theta^{L-F+multip1icationtrans}}\Leftrightarrow.\theta L_{\psi}^{-1}T_{\psi,H,\lambda}-1.$

We compare changes of the spectrum of the operators
as changes form to 1. For details, see [18].
$1$$\theta$

4.2$D_{-\lambda}-A_{\lambda,\psi,\theta}$

and$\theta L_{\psi}^{-1}T_{\psi,H,\lambda}-$$0$

Definition of$\square$$m_{\mathbb{R}^{m}}$Motivated by the formula of Theorem 6, for$\psi\in L^{\infty}(M, \mathbb{S}(M))$, we define$T_{\psi}=|\psi|^{R_{\frac{-1}{2}}}\circ D_{g_{\mathbb{R}^{m}}}^{-1}\circ|\psi|^{g_{\frac{-1}{2}}},$. is the Green kernel ofwhere
The integral representation ofis given as$D_{g_{\mathbb{R}^{m}}}^{-1}=-\omega_{m-1}^{-1}\frac{(x-y)}{|x-y|^{m}}$$D_{\mathfrak{W}^{m}}.$$T_{\psi}$$(T_{\psi} \varphi)(x)=-\frac{1}{\omega_{m-1}}|\psi|^{L_{\frac{-1}{2}}}(x)\int_{\mathbb{R}^{m}}\frac{(x-y)}{|x-y|^{m}}$Note that:
ator. We list below basic properties of$T_{\psi}$$(|\psi|^{g}2\llcorner-\underline{1}(y)\varphi(y))dvo1_{g_{\mathbb{R}^{m}}}(y)$.

is in general an unbounded oper. For the proof, see [18].

$L^{2}(\mathbb{R}^{m}, \mathbb{S}(\mathbb{R}^{m}))arrow L^{2}(\mathbb{R}^{m}, \mathbb{S}(\mathbb{R}^{m}))$$T_{\psi}$(1).

$T_{\psi}$

is a densely defined self-adjoint operator on(2) Its domain$\mathcal{D}(T_{\psi})$contains$L^{\infty}$$L^{2}(\mathbb{R}^{m})$when $m\geq 3.$-spinors with compact supports when $m\geq 3.$

Definition 3 Let
be a non-trivial solution toon. We define relative Morse index
as the dimension of the maximal
subspace ofon which the following holds$\psi\in L^{\infty}(\mathbb{R}^{m}, \mathbb{S}(\mathbb{R}^{m}))$$D_{g_{\mathbb{R}^{m}}}\psi=|\psi|^{p-1}\psi$$\mathbb{R}^{m}$$m_{\mathbb{R}^{m}}(\psi)$$\mathcal{D}(T_{\psi})$$\frac{(T_{\psi}(\varphi),\varphi)_{L^{2}(\mathbb{R}^{m})}}{(L_{\psi}(\varphi),\varphi)_{L^{2}(\mathbb{R}^{m})}}>1.$

With this definition, properties ( -1), ( -2) in \S 3.4 are consequences of Theorem 6.
The proof of ( -3) requires a uniform estimate of the Schwartz kernel ofalong
the blowing-up manifolds. For details, see [18].
$I$

$I$

$I$

$D_{g_{n}}^{-1}$$(M, g_{n})arrow(\mathbb{R}^{m}, g_{\mathbb{R}^{m}})$4.3

Outline of the proof of ( -4)(Theorem 4)$I$

The proof of ( -4) (Theorem 4) is based on the following:$I$

Theorem 7 Assume that $m\geq 2$ and $1<p< \frac{m+1}{m-1}$ . Let
solution to. Then we haveon
$D_{g_{\mathbb{R}^{m}}}\psi=|\psi|^{p-1}\psi$$\mathbb{R}^{m}$$\psi\in L^{p+1}(\mathbb{R}^{m}, \mathbb{S}(\mathbb{R}^{m}))$$\psi\equiv 0.$

be a

90
Sketch of the proof of ( -4): We observe that$I$$d^{2} \mathcal{L}(\psi)(\psi, \psi)=(1-p)\int_{\mathbb{R}^{m}}|\psi|^{p+1}dvo1_{g_{R^{m}}}<0$

for$\mathcal{L}(\varphi)=\frac{1}{2}\int_{\mathbb{R}^{m}}\langle\varphi,$$D_{g_{R^{m}}}\varphi\rangle dvo1_{g_{R^{m}}}-\frac{1}{p+1}\int_{\mathbb{R}^{m}}|\psi|^{p+1}dvo1_{g_{R^{m}}}$. (However, this cal-culation is not correct by Theorem 7!). By this (incorrect) calculation, a candidate
with eigenvalue larger than 1 is given by
for eigenspinor of. In fact,, i.e.,
multiplied by$D_{g_{R^{m}}}\psi=|\psi|^{p-1}\psi$$L_{\psi}^{-1}T_{\psi}$$\varphi=|\psi|^{L_{2}^{-\underline{1}}}\psi$$|\psi|^{-2_{\frac{-1}{2}}}$$(T_{\psi}( \varphi), \varphi)_{L^{2}}=\int_{\mathbb{R}^{m}}|\psi|^{p+1}dvo1_{g_{R^{m}}},$$(L_{\psi}( \varphi), \varphi)_{L^{2}}=\frac{1}{p}\int_{\mathbb{R}^{m}}|\psi|^{p+1}dvo1_{g_{R^{m}}}$a 皿 $d$$\frac{(T_{\psi}(\varphi),\varphi)_{L^{2}}}{(L_{\psi}(\varphi),\varphi)_{L^{2}}}=p>1.$(But, the calculation is not correct.)
To give a correct proof, we truncate

be such thatsuitably: Let$\eta(x)=1$ for $|x|\leq 1$ and $\eta(x)=0$ for $|x|\geq 2$ . For $R>0$ , define $\eta_{R}(x)=\eta(x/R)$

and$\eta\in C_{0}^{\infty}(\mathbb{R}^{m})$$\varphi$$\varphi_{\ell,R}=\eta_{R}^{\ell}|\psi|^{L_{2}^{\underline{-1}}}\psi.$

For large$R$and large ($\ell$$\ell\geq L_{\frac{1}{1}}^{+}p-$is sufficient) , we can show that (see [18] for details)$\frac{(T_{\psi}(\varphi_{\ell,R}),\varphi_{\ell,R})_{L^{2}}}{(L_{\psi}(\varphi_{\ell,R}),\varphi_{\ell,R})_{L^{2}}}>1.$

survives at infinity and suitably chosen sequence of cut-offs $R_{1}<$such that$R_{2}<\cdots<R_{\eta}<\cdots$ give arbitrary number of linearly independent

By Theorem 7,$\psi$$\varphi_{\ell,R_{j}}$$\frac{(T_{\psi}(\varphi_{\ell,R_{j}}),\varphi_{\ell,R_{j}})_{L^{2}}}{(L_{\psi}(\varphi_{\ell,R_{j}}),\varphi_{\ell,R_{j}})_{L^{2}}}>1.$

The asSertion of Theorem$4f\circ 11_{oWS}$

from this.

口

91
5

Morse-Floer homology$HF_{*}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M))$

In a recent work, Maalaoui [20] constructed Rabinowitz-Floer homology ([11]) for
Dirac equations. It is a Floer homology for Lagrangian multiplier functional andit may be considered as a way of defining Floer homology for pure spinor action
functional
on manifold $\{\psi : \int_{M}H(x, \psi)=1\}$ . However, for thepresent author, some of his arguments and assertions are difficult to understand. It
seems that some more additional arguments are necessary to verify his assertions.
In any way, we consider “free” action functional and give an outline of the construction and the computation of the Morse-Floer homology of
associatedtounder assuming (2.2) and (2.3). In fact, it is not necessary to assume (2.2).
We only need weaker condition$\int_{M}\langle\psi,$$D_{g}\psi\rangle dvo1_{g}$$\mathcal{H}^{1/2}(M)$$\mathcal{L}_{H}$$|H_{\psi}(x, \psi)|\leq C(1+|\psi|^{p})$

.

(5.1)

Assumeis Morse. This is a generic condition for $H$ , see [18] for the proof. We
also assume
for simplicity. The casewill be treated in [18].
Recall that the graded groupis defined as$\mathcal{L}_{H}$$\mathbb{F}=\mathbb{Z}$$\mathbb{F}=\mathbb{Z}_{2}$$\{C_{p}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M))\}_{p\in \mathbb{Z}}$$C_{p}( \mathcal{L}_{H}, \mathcal{H}^{1/2}(M))=\bigoplus_{\psi\in crit_{p}(\mathcal{L}_{H})}\mathbb{Z}_{2}\langle\psi\rangle,$

where$crit_{p}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M))=\{\psi\in$crit$(\mathcal{L}_{H})$: $m_{\lambda}(\psi)=p\}.$

We next give the definition of the boundary operator:$:=\{\psi\in \mathcal{H}^{1/2}(M) : d\mathcal{L}_{H}(\psi)=0\}$. Let x, y crit. Let
be such thatfor $t\leq-1,$ $\psi_{0}(t)=y$ for $t\geq 1$ . Define
the trajectory space connecting and by$C_{p}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M))arrow$$\partial_{p}$$C_{p-1}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M))$$(\mathcal{L}_{H})$$\in$$\psi_{0}\in C^{1}(\mathbb{R}, \mathcal{H}^{1/2}(M))$$\psi_{0}(t)=\cross$

$x$

$y$

$M(x, y)=\{\psi\in\psi_{0}+W^{1,2}(\mathbb{R}, \mathcal{H}^{1/2}(M)):\frac{\partial\psi}{\partial t}=-\nabla_{1/2}\mathcal{L}_{H}(\psi)$

Note that,$\psi(-\infty)=x,$ $\psi(+\infty)=y\}.$acts freely on $M(x, y)$ via the time shift and we obtain the moduli spaceof unparametrized trajectories connecting and$M(x, y)$ and hence
are manifolds if is a regular value of Fredholm map$\mathbb{R}$$\hat{M}(x, y)=M(x, y)/\mathbb{R}$$x$$\hat{M}(x, y)$

$y.$

$0$

.

$\mathcal{F}_{H}:W^{1,2}(\mathbb{R}, \mathcal{H}^{1/2}(M))\ni\psi\mapsto\frac{\partial\psi}{\partial t}+\nabla \mathcal{L}_{H}(\psi)\in L^{2}(\mathbb{R}, \mathcal{H}^{1/2}(M))$

This is equivalent to the condition that $W^{u}(x)$ and $W^{S}(y)$ intersect transversally atfor some(and hence for all), where the unstable manifold $W^{u}(x)$and the stable manifold $W^{S}(y)$ are defined by$\psi(t)$$t\in \mathbb{R}$$t\in \mathbb{R}$$W^{u}( \cross)=\{z\in \mathcal{H}^{1/2}(M):\lim_{tarrow\infty}\psi(t, z)=x\},$whereis the solution to
Recall that the negative gradient flowintersect transversally for any pair x, y crit$\psi(t, z)$$W^{s}( y)=\{z\in \mathcal{H}^{1/2}(M):\lim_{tarrow+\infty}\psi(t, z)=y\},$$\partial A\partial t+\nabla \mathcal{L}_{H}(\psi)=0,$$\psi(t, \cdot)$$\in$$\psi(0, z)=z.$

is Morse-Smale if $W^{u}x$ ) and $W^{u}(y)$. In this case,is a manifold$(\mathcal{L}_{H})$$\hat{M}(x, y)$92
is Morse-Smale onof $\dim\hat{M}(x, y)=m_{\lambda}(x)-m_{\lambda}(y)-1$ . We assume that(As we will explain shortly, this condition is in general never satisfied for$\mathcal{H}^{1/2}(M)$$\mathcal{L}_{H}$, however).
Under the assumption, we have:$\dim\hat{M}(x, y)=0$ when
Furthermore, if
is compact,$\mathcal{L}_{H}$.

on$\mathcal{H}^{1/2}(M)$$x\in crit_{p}(\mathcal{L}_{H}),$$\hat{M}(x, y)$$\hat{M}$.
(x, y) consists of a finite number of points and$y\in crit_{p-1}(\mathcal{L}_{H})$

we define$\partial_{p^{\cross=}}\sum_{y\in crit_{p-1}(L_{H})}n(x, y)y$

(5.2)

,

where $n(x, y)=\#(\hat{M}(x, y))(mod 2)$

To prove the boundary property
with $m_{\lambda}(x)-m_{\lambda}(y)=2$ and prove$\partial_{p}\partial_{p-1}=0$, we need$\hat{M}(\cross, z)$

for x, z

$\in$

crit

$(\mathcal{L}_{H})$

.

(5.3)

$\partial_{p}\partial_{p-1\cross=}\sum_{z\in crit_{p-2}(\mathcal{L}_{H})}\sum_{y\in crit_{p-2}(\mathcal{L}_{H})}n(x, y)n(y, z)z=0$

Thus, we want $M(x, y)$ to be manifold for x, y

$\in$

crit

$(\mathcal{L}_{H})$with $m_{\lambda}(x)-m\lambda(y)=1$

and 2.
To obtain Morse-Smale property for generic $H$ , there are some technical problems
which are not present in 1-dimensional variational problems. (Hamiltonian systems
on symplectic manifolds are typical ones). More precisely, we have the following
problems to construct the Morse-Floer homology for our case:

Regularity: We wantis Morse for a generic $H$ . We also want for a generic$H$ and a generic metric on, gradient flows are Morse-Smale at least forx, y critwith$\mathcal{L}_{H}$$\bullet$$\mathcal{H}^{1/2}(M)$$\in$$(\mathcal{L}_{H})$$m_{\lambda}(\cross)-m_{\lambda}(y)\leq 2.$

Compactness: We wantcation.

$\hat{M}(x, y)$to be precompact and has a natural compactifi-

However, they conflict to each other:
It is possible to prove thatis Morse for generic $H$ . To obtain the Morse-Smale
property as stated above, we need at least. This is-regularity foron
the regularity versus Fredholm index assumption required for the use of Sard-Smale
theorem. However,is at mostoneven if we assume. To
remedy the lack of regularity which occurs when we are working on, weneed to work on more regular spinor space.
On the other hand, compactness is easier to obtain when working on less regularspinor space like.
$\mathcal{L}_{H}$$\bullet$$C^{3}$$\mathcal{L}_{H}$$C^{2}$$\mathcal{H}^{1/2}(M)$$\mathcal{L}_{H}$$\mathcal{H}^{1/2}(M)$$H\in C^{\infty}$$\mathcal{H}^{1/2}(M)$$\bullet$$\mathcal{H}^{1/2}(M)$

Abbondandolo and Majer [2], [3], [4] constructed general Morse-Floer theory fora class of strongly indefinite functional defined on Hilbert manifolds. However,their general theory does not directly applicable due to the above problems. In afundamental work of [6], they constructed Morse-Floer homology for elliptic systems.

93
In that problem, similar problems also raised. Our construction of the Morse-Floer
homology of
associated toowes much to their ideas.
Under subcritical and superquadratic conditions on $H$ , together with the ellipticity ofand its nice mapping properties in various function spaces, insteadof working with, we can work with more regular spinor space$:=$for $1/2<\alpha<1)$ and resolve both problems at the same time. Thatis, we have:$\mathcal{H}^{1/2}(M)$$\mathcal{L}_{H}$$D_{g}$$\mathcal{H}^{1/2}(M)$$C^{0,\alpha}(M, \mathbb{S}(M))$

(1)

is

$\mathcal{L}_{H}$(2) crit$C^{k}$$($

on$\mathcal{C}^{0,\alpha}(M)(\frac{1}{2}<\alpha<1)$$(\mathcal{L}_{H})\subset \mathcal{C}^{0,\alpha}(M)$(3)$\mathcal{C}^{0,\alpha}(M)$(4)$\mathcal{L}_{H}$$\mathcal{C}^{0,\alpha}(M)$

if $H$ is $C^{k+1}.$:

is invariant under the$H^{1/2}$-gradient flow.

satisfies the Palais-Smale condition on(5) $\psi\in M(x, y)$ is uniformly bounded in$\mathcal{H}^{1/2}(M)$.

$\mathcal{C}^{0,\alpha}(M);\sup_{t\in \mathbb{R}}\Vert\psi(t)\Vert_{C^{0,\alpha}}<+\infty.$(1) $-(3)$ are regularity conditions, while (4) and (5) are compactness conditions.

There are also technical issues to overcome. In anyway, from these, we have (see [18]for details)(1)$\mathcal{L}_{H}$(2) Foris Morse for generic $H.$

critwith $m_{\lambda}(x)-m_{\lambda}(y)\leq 2,\hat{M}(x, y)$ is a manifold of dimension$m_{\lambda}(x)-m\lambda(y)-1$ for generic $H$ and generic metric on.
$\cross,$$y\in$$(\mathcal{L}_{H})$$\mathcal{H}^{1/2}(M)$$\frac{(3}{\hat{M}}(\cross,y)f\in crit(\mathcal{L}_{H})withm(\cross)-m(y)\leq 2)\hat{M}(x, y)\subset C_{1oc}^{0}(\mathbb{R}, \mathcal{C}^{0,\alpha}(M))irecom_{\lambda}$

has a natural compactification.
Also, it is easy to see thatis a gradient flow invariant compact set. Thus, if
is Morse and the gradient flow is Morse-Smale, thenconsists of a finitely
many critical points ofand connecting orbits of these. From these, we have:$\partial\hat{M}(x, y)-$$\hat{M}(x, y)$$\mathcal{L}_{H}$$\mathcal{L}_{H}$is compact when $m(x)-m(y)=1$ anddefined by the formula (5.2) is well-defined.
$\bullet$$\hat{M}(x, y)$$\partial_{p}$:$C_{p}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M))arrow C_{p}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M))$

is a 1-dimentional manifold with boundary when $m_{\lambda}(x)-m\lambda(z)=2.$
consists precisely of 1-breaking” orbits (this requires gluing construction
which we skipped in the above argument).
$\bullet$$J\hat{v}[(\cross, z)$$\partial\hat{M}(x, z)$

In the formula of$\partial_{p-1}\partial_{p}$in (5.3), the matrix element$\sum_{y\in crit_{p-1}(\mathcal{L}_{H})}n(x, y)n(y, z)$counts the number of the connected components of$(mod 2)$ . This proves the boundary property:
well-defined homology of the chain complex

which is even and. In this way, we have a$\partial\hat{M}(x, z)$$\partial_{p-1}\partial_{p}=0$$\{C_{p}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M)), \partial_{p}\}_{p\in \mathbb{Z}}$$HF_{p}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M);\mathbb{Z}_{2}) :=ker\partial_{p}/{\rm Im}\partial_{p+1}$

$0$

:

94
.
for generic $H$ and generic metric on
To definefor general $H$ , we take a generic $H’$ which satisfies (2.1), (2.2) andand a generic metric on
such
that
is Morse and the negative gradient flow system is Morse-Smale. Thenis defined (we omitted to indicate the dependence of the, butmetric on
defined so far indeed depends on.
the choice of the metric on
The next step is to show that
does not depend on the$H’$choices of a generic. Note that, in general, theand a generic metric on
Morse homology on a non-compact manifold depends on a chosen Morse function.
This is in contrast to the compact case. Thus, to obtain stability result, we need ageneral result is stated as follows:
restriction on a class of functions.
$\mathcal{H}^{1/2}(M)$$HF(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M);\mathbb{Z}_{2})$$\mathcal{H}^{1/2}(M)$$\Vert H-H’\Vert_{L\infty}<\epsilon$$\mathcal{L}_{H’}$$HF_{*}(\mathcal{L}_{H’}, \mathcal{H}^{1/2}(M);\mathbb{Z}_{2})$$\mathcal{H}^{1/2}(M)$$HF_{*}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M);\mathbb{Z}_{2})$$\mathcal{H}^{1/2}(M))$$HF_{*}(\mathcal{L}_{H’}, \mathcal{H}^{1/2}(M);\mathbb{Z}_{2})$$\mathcal{H}^{1/2}(M)$$A$

Theorem 8 For generic pairs
satisfying (2.1), (2.2) andnatural isomorphism$H_{0},$$G_{0},$ $G_{1}$$G_{1}(H_{0},$andare generic metrics on$G_{0}$$H_{1},$are generic functions, we have a$H_{1}$$\mathcal{H}^{1/2}(M))$$HF_{*}(\mathcal{L}_{H_{0}}, (\mathcal{H}^{1/2}(M), G_{0});\mathbb{Z}_{2})\cong HF_{*}(\mathcal{L}_{H_{1}}, (\mathcal{H}^{1/2}(M), G_{1});\mathbb{Z}_{2})$provided$\Vert H_{0}-H_{1}\Vert_{L(\mathbb{S}(M))}\infty<+\infty.$

By Theorem 8, we can define$HF_{*}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M))=HF_{*}(\mathcal{L}_{H’}, (\mathcal{H}^{1/2}(M), G’))$for a generic $H’$ and $G’$ which satisfies $\Vert H-H’\Vert_{L(S(M)}\infty<+\infty.$

Outline of the proof of Theorem 8: The proof is also standard, see [13], [6]. Letbe such that $\rho(t)=0$ for $t\leq-1$ and $\rho(t)=1$ for $t\geq 1$ . We consider
time dependent function and metric which interpolate between
andand, respectively:
and$\rho\in C^{\infty}(\mathbb{R})$$H_{0}$$H_{1}$$G_{0}$$G_{1}$$H_{1,0}(t, x, \psi)=(1-\rho(t))H_{0}(x, \psi)+\rho(t)H_{1}(x, \psi)$$G_{1,0}(t, \psi)=(1-p(t))G_{0}(\psi)+\rho(t)G_{1}(\psi)$,.

We consider non-autonomous system$\frac{\partial\psi}{\partial t}=-\nabla_{G_{1,0}}\mathcal{L}_{H_{1,0}}(\psi) ,\psi(-\infty)=\cross 0, \psi(+\infty)=x_{1},$

where
with
and
is the gradient ofrespect to the metric.
We consider the moduli space of solutions to the above system
Under the assumption,satisfies (2.1) and (2.2) uniformly for
and we canshow, as in the autonomous case,is precompact inand it has a natural compactification whose boundary consists of broken trajectories.
$x_{0}\in crit_{p}(\mathcal{L}_{H_{0}}),$ $x_{1}\in crit_{q}(\mathcal{L}_{H_{1}})$$\nabla_{G_{1,0}}\mathcal{L}_{H_{1,0}}$$\mathcal{L}_{H_{1,0}}$$G_{1,0}.$$M_{H_{1,0},G_{1,0}}(x_{0}, x_{1})$$t\in \mathbb{R}$$H_{1,0}$$M_{H_{1,0},G_{1,0}}(x_{0}, x_{1})$$C_{1oc}^{0}(\mathbb{R}, \mathcal{C}^{0,\alpha}(M))$

95
After perturbing
and
if necessary,sion. In particular, for the caseis compact and we can define$H_{1,0}$$G_{1,0}$$M_{H_{1,0},G_{1,0}}(x_{0}, x_{1})$$m_{\lambda}(x_{0})-m_{\lambda}(x_{1})$is a manifold of dimen-$m_{\lambda}(x_{0})=m_{\lambda}(x_{1}),$$M_{H_{1,0},G_{1,0}}(\cross 0, x_{1})$$\Phi:C_{p}(\mathcal{L}_{H_{0}}, \mathcal{H}^{1/2}(M))arrow C_{p}(\mathcal{L}_{H_{1}}, \mathcal{H}^{1/2}(M))$

by counting trajectories:$\Phi_{1,0}(x_{0})=\sum_{x_{1}\in crit_{p}(\mathcal{L}_{H_{1}})}n_{H_{1,0},G_{1,0}}(x_{0}, x_{1})x_{1},$where $n_{H_{1,0},G_{1,0}}(x_{0}, x_{1})=\#M_{H_{1,0},G_{1,0}}(x_{0}, x_{1})(mod 2)$ .
By examining the boundary
for
crit
and
crit
with
, we see thatis a chain map:
. Moreover, considering homotopies of homotopies, we see thatis natural in the sense that$\partial M_{H_{1,0},G_{1,0}}(x_{0}, x_{1})$$m_{\lambda}(\cross 0)-m_{\lambda}(x_{1})=1$$(\mathcal{L}_{H_{1}})$$x_{0}\in$$\Phi_{1,0}$$(\mathcal{L}_{H_{0}})$$x_{1}\in$$\partial_{H_{1}}\circ\Phi_{1,0}+$$\Phi_{1,0}\circ\partial_{H_{0}}=0$$\Phi_{1,0}$$\Phi_{0,0}=1, \Phi_{2,1}\circ\Phi_{1,0}=\Phi_{2,0}.$These imply that$\Phi_{1,0}$

induces the isomorphism of homologies:$\Phi_{1,0}:HF_{*}(\mathcal{L}_{H_{0}}, (\mathcal{H}^{1/2}(M), G_{0}))\cong HF_{*}(\mathcal{L}_{H_{1}}, (\mathcal{H}^{1/2}(M), G_{1}))$.
ロ

Based on the above Theorem 8, we have
Theorem 9 Assume that $H\in C^{2}(\mathbb{S}(M))$ satisfies (2.1) and (2.2). Then the MorseFloer homology
is well-defined. For the case $H(x, \psi)=$(more generally, for $H$ satisfying “strong superquadratic conditionat \‘infinity’’), we have a vanishing result$FH_{*}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M);\mathbb{Z}_{2})$$\frac{1}{p+1}H(x)|\psi|^{p+1}$$HF_{*}(\mathcal{L}_{H}, \mathcal{H}^{1/2}(M))=0.$

For details, see [18].

参考文献[1] A. Abbondandolo: Morse theory for Hamiltonian systems. CRC Press, (2001).

[2] A. Abbondandolo, P. Majer: Morse homology on Hilbert spaces. Comm. Pureand Appl. Math. 54 (2001), 689-760.
[3] A. Abbondandolo, P. Majer: “Lectures on the Morse complex for infinitedimensional manifolds.” Morse theoretic methods in nonlinear analysis and in
symplectic topology. Springer Netherlands (2006), 1-74.

[4] A. Abbondandolo, P. Majer:Ordinary differential operators in Hilbert spaces
and Fredholm pairs. Math. Z. 243 (2003), 525-562.

96[5] A. Abbondandolo, P. Majer: A Morse complex for infinite dimensional manifolds–part I. Adv. Math. 197 (2005), 321-410.

superquadratic indefinite elliptic system and[6] S. Angenent, R. van der Vorst:
its Morse-Conley-Floer homology. Math. Z. 231 (1999), 203-248.
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17 (2000), 277-306.
[8] A. Bahri, P. L. Lions: Solutions of superlinear elliptic equations and their Morse
indices. Comm. Pure and Appl. Math., 45 (1992), 1205-1215.

[9] Q. Chen, J. Jost, G. Wang: Liouville theorems for Dirac-harmonic maps. J.
Math. Phys 48 (2007), 113517.
[10] Q. Chen, J. Jost, J. Li, G. Wang: Dirac-harmonic maps. Math. Z. 254.2 (2006):

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