Morse homology for the heat flow

Abstract

We use the heat flow on the loop space of a closed Riemannian manifold—viewed as a parabolic boundary value problem for infinite cylinders—to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined by counting, modulo time shift, heat flow trajectories between geodesics of Morse index difference one. By Salamon and Weber (GAFA 16:1050–138, 2006) this heat flow homology is naturally isomorphic to Floer homology of the cotangent bundle for Hamiltonians given by kinetic plus potential energy.

Appendix A: Parabolic regularity

By $\mathbb{H }^-$ we denote the closed lower half plane, that is, the set of pairs of reals $(s,t)$ with $s\le 0$. For now all maps are real-valued and the domains of the various Banach spaces which appear are understood to be open subsets $\varOmega$ of $\mathbb{R }^2$ or $\mathbb{H }^-$. To deal with the heat equation it is useful to consider the anisotropic Sobolev spaces $W_p^{k,2k}$. We call them parabolic Sobolev spaces and denote them by $\mathcal{W }^{k,p}$. For constants $p\ge 1$ and integers $k\ge 0$ these spaces are defined as follows. Set $\mathcal{W }^{0,p}=L^p$ and denote by $\mathcal{W }^{1,p}$ the set of all $u\in L^p$ which admit weak derivatives ${\partial }_su$, ${\partial }_tu$, and ${\partial }_t{\partial }_tu$ in $L^p$. For $k\ge 2$ define

$$\begin{aligned} \mathcal{W }^{k,p}:=\{ u\in \mathcal{W }^{1,p}\mid {\partial }_su,{\partial }_tu,{\partial }_t{\partial }_tu\in \mathcal{W }^{k-1,p}\} \end{aligned}$$

where the derivatives are again meant in the weak sense. The norm

$$\begin{aligned} \left\Vert {u} \right\Vert_{\mathcal{W }^{k,p}} :=\left( \int \int \sum _{2\nu +\mu \le 2k} \left|{\partial }_s^\nu {\partial }_t^\mu u(s,t)\right|^p\, dt ds \right)^{1/p} \end{aligned}$$

(79)

gives $\mathcal{W }^{k,p}$ the structure of a Banach space. Here $\nu$ and $\mu$ are nonnegative integers. For $k=1$ we obtain that

$$\begin{aligned} \left\Vert {u} \right\Vert_{\mathcal{W }^{1,p}}^p =\left\Vert {u} \right\Vert_p^p +\left\Vert {{\partial }_su} \right\Vert_p^p +\left\Vert {{\partial }_tu} \right\Vert_p^p +\left\Vert {{\partial }_t{\partial }_tu} \right\Vert_p^p \end{aligned}$$

and occasionally we abbreviate $\mathcal{W }=\mathcal{W }^{1,p}$. Note the difference to (standard) Sobolev space $W^{k,p}$ where the norm is given by $\mathopen \Vert {u} \mathclose \Vert _{k,p}^p :=\sum _{\nu +\mu \le k} \mathopen \Vert {{\partial }_s^\nu {\partial }_t^\mu u} \mathclose \Vert _p^p$. A rectangular domain is a set of the form $I\times J$ where $I$ and $J$ are bounded intervals. For rectangular (more generally, Lipschitz) domains $\varOmega$ the parabolic Sobolev spaces $\mathcal{W }^{k,p}$ can be identified with the closure of $C^\infty (\overline{\varOmega })$ with respect to the $\mathcal{W }^{k,p}$ norm; see e.g. [8, app. B.1]. Similarly, define the $\mathcal{C }^k$ norm by

$$\begin{aligned} \left\Vert {u} \right\Vert_{\mathcal{C }^k} :=\sum _{2\nu +\mu \le 2k} \left\Vert {{\partial }_s^\nu {\partial }_t^\mu u} \right\Vert_{\infty }. \end{aligned}$$

(80)

Assume $N\hookrightarrow \mathbb{R }^N$ is a closed smooth submanifold and $\varGamma :M\rightarrow \mathbb{R }^{N\times N\times N}$ is a smooth family of vector-valued symmetric bilinear forms. Set $\mathcal{W }^{k,p}(Z)=\mathcal{W }^{k,p}(Z,\mathbb{R }^N)$ and for $T>T^\prime >0$ set $Z=Z_T=(-T,0]\times S^1$ and $Z^\prime =Z_{T^\prime }$.

Proposition 12

(Parabolic regularity) Fix constants $p>2$, $\mu _0>1$, and $T>0$. Fix a map $F:Z\rightarrow \mathbb{R }^N$ such that $F$ and ${\partial }_tF$ are of class $L^p$. Assume that $u:Z\rightarrow \mathbb{R }^N$ is a $\mathcal{W }^{1,p}$ map taking values in $N$ with $\mathopen \Vert {u} \mathclose \Vert _{\mathcal{W }^{1,p}}\le \mu _0$ and such that the perturbed heat equation

$$\begin{aligned} {\partial }_su-{\partial }_t{\partial }_tu =\varGamma (u)\left({\partial }_tu,{\partial }_tu\right) +F \end{aligned}$$

(81)

is satisfied almost everywhere. Then the following is true for every integer $k\ge 1$ such that $F,{\partial }_tF\in \mathcal{W }^{k-1,p}(Z)$ and every $T^\prime \in (0,T)$.

• (i) There is a constant $a_k$ depending on $p$, $\mu _0$, $T$, $T^\prime$, $\mathopen \Vert {\varGamma } \mathclose \Vert _{C^{2k+2}}$, and the $\mathcal{W }^{k-1,p}(Z)$ norms of $F$ and ${\partial }_tF$ such that

  $$\begin{aligned} \left\Vert {{\partial }_tu} \right\Vert_{\mathcal{W }^{k,p}(Z^\prime )} \le a_k. \end{aligned}$$

• (ii) If ${\partial }_sF\in \mathcal{W }^{k-1,p}(Z)$ then there is a constant $b_k$ depending on $p$, $\mu _0$, $T$, $T^\prime$, $\mathopen \Vert {\varGamma } \mathclose \Vert _{C^{2k+2}}$, and the $\mathcal{W }^{k-1,p}(Z)$ norms of $F$, ${\partial }_tF$, and ${\partial }_sF$ such that

  $$\begin{aligned} \left\Vert {{\partial }_su} \right\Vert_{\mathcal{W }^{k,p}(Z^\prime )} \le b_k. \end{aligned}$$

• (iii) If ${\partial }_t{\partial }_tF\in \mathcal{W }^{k-1,p}(Z)$ then there is a constant $c_k$ depending on $p$, $\mu _0$, $T$, $T^\prime$, $\mathopen \Vert {\varGamma } \mathclose \Vert _{C^{2k+2}}$, and the $\mathcal{W }^{k-1,p}(Z)$ norms of $F$, ${\partial }_tF$, and ${\partial }_t{\partial }_tF$ such that

  $$\begin{aligned} \left\Vert {{\partial }_t{\partial }_tu} \right\Vert_{\mathcal{W }^{k,p}(Z^\prime )} \le c_k. \end{aligned}$$

Since $p>2$, the Sobolev embedding theorem guarantees that every $\mathcal{W }^{1,p}$ map $u$ is continuous. Hence it makes sense to say that $u$ takes values in $N$.

Corollary 2

Under the assumptions of Proposition 12 the following is true. Assume $k\ge 1$ and $F\in \mathcal{W }^{k,p}(Z_T)$. Then for every $T^\prime \in (0,T)$ there is a constant $c_k=c_k(k,p,\mu _0,T-T^\prime , \mathopen \Vert {\varGamma } \mathclose \Vert _{C^{2k+2}(N)}, \mathopen \Vert {F} \mathclose \Vert _{\mathcal{W }^{k,p}(Z_T)})$ such that

$$\begin{aligned} \left\Vert {u} \right\Vert_{\mathcal{W }^{k+1,p}(Z_{T^\prime })} \le c_k. \end{aligned}$$

Proof

The $\mathcal{W }^{k+1,p}$ norm of $u$ is equivalent to the sum of the $\mathcal{W }^{k,p}$ norms of $u$, ${\partial }_tu$, ${\partial }_su$, and ${\partial }_t{\partial }_tu$. Apply Proposition 12 (i–iii). $\square$