Convergence Stability for Ricci Flow

Abstract

The principle of convergence stability for geometric flows is the combination of the continuous dependence of the flow on initial conditions, with the stability of fixed points. It implies that if the flow from an initial state $g_0$ exists for all time and converges to a stable fixed point, then the flows of solutions that start near $g_0$ also converge to fixed points. We show this in the case of the Ricci flow, carefully proving the continuous dependence on initial conditions. Symmetry assumptions on initial geometries are often made to simplify geometric flow equations. As an application of our results, we extend known convergence results to open sets of these initial data, which contain geometries with no symmetries.

Appendix A. Geometric Background

For the convenience of the reader, in this appendix we review notation, define appropriate function spaces and review key facts from elliptic PDE theory. Expanded summaries can be found in [7, 22].

Let E be a smooth tangent tensor bundle over a compact Riemannian manifold $(\mathcal {M}^m,g)$. For smooth sections u, v, we define the $L^2$ pairing by

$$\begin{aligned} (u, v) := \int _{\mathcal {M}} \left\langle u, v \right\rangle _g {{\,\mathrm{dvol}\,}}_g, \end{aligned}$$

and we set $\Vert u \Vert _{L^2}^2 := (u,u)$. Let $C^{\infty }(E)$ denote the space of smooth sections of E, and define $L^2(E)$ as the completion of smooth tensor fields with respect to the $L^2$ norm. In a similar way, for any $k \in \mathbb {N}_0$ and for any $p \in [1,\infty )$, we introduce the Sobolev space of tensor fields with k covariant derivatives in $L^p$ by taking the completion of $C^{\infty }(E)$ with respect to the norm

$$\begin{aligned} \Vert u \Vert _{L^{k,p}}^p := \sum _{i=0}^k \int _{\mathcal {M}} \Vert \nabla ^{(i)} u \Vert ^p_g {{\,\mathrm{dvol}\,}}_g, \end{aligned}$$

where $\nabla$ denotes the covariant derivative with respect to the background metric g.

We also need Hölder spaces. Consider any finite covering of $(\mathcal {M},g)$ by open balls $\mathscr {C} = \{(B_i, \phi _i)\}_{i=1}^N$ with charts $\phi _i: B_i \subset \mathcal {M}\rightarrow \phi (B_i) \subset \mathbb {R}^m$ and let $\psi _i$ be an associated smooth partition of unity, so that $\mathrm {supp}(\psi _i) \subset B_i$ and $\sum _{i=1}^N \psi _i = 1$. Now if u is a continuous real or complex-valued function on $\mathcal {M}$, recall that the sup-norm of u is defined by

$$\begin{aligned} \Vert u\Vert _{\infty } := \sup _{q \in \mathcal {M}} |u(q)|. \end{aligned}$$

Note that if u is entirely supported inside one coordinate chart $(B_i, \phi _i) \in \mathscr {C}$, then we may define the Hölder quotient of u by simply computing the Hölder quotient of the coordinate representation of u with respect to a background Euclidean metric, $\delta$; in other words,

$$\begin{aligned}{}[u]_{\alpha ; B_i} := \sup _{x \ne x', x,x' \in \phi (B_i)} \frac{ \left| u(\phi _i^{-1}(x)) - u(\phi _i^{-1}(x')) \right| }{|x-x'|^{\alpha }}. \end{aligned}$$

(24)

For functions not supported in a single coordinate chart, let us define a Hölder norm with respect to a covering $\mathscr {C}$ by

$$\begin{aligned} \Vert u\Vert _{C^{0,\alpha }; \mathscr {C}} := \Vert u\Vert _{\infty } + \sum _{i=1}^N {[}\psi _i u]_{\alpha ; B_i}, \end{aligned}$$

and then define the space $C^{0,\alpha }(\mathcal {M})$ as the set of continuous functions on $\mathcal {M}$ such that $\Vert u\Vert _{C^{0,\alpha };\mathscr {C}} < \infty$. While this definition depends on the choice of cover, one may check that membership in $C^{0,\alpha }$ is independent of this choice and that these norms are all equivalent up to constants depending on the cover. We hereafter omit the symbol $\mathscr {C}$ from the norm notation.

Regarding Hölder spaces for tensor fields, the key change needed is that in the local definition of the Hölder space for functions (see (24)), we replace $u(\phi _i^{-1}(x))$ by $(\phi ^{-1}_i)^* u(x)$, and then compute the Hölder quotient of the Euclidean norm of the components of u in coordinates as follows:

$$\begin{aligned} {[}u]_{\alpha ; B_i} := \sup _{x \ne x', x,x' \in \phi _i(B_i)} \frac{ \left| (\phi _i^{-1})^* u (x) - (\phi _i^{-1})^* u(x') \right| _{\delta }}{|x-x'|^{\alpha }}. \end{aligned}$$

(25)

Using this definition for the Hölder norm on a patch $B_i$, we define the $C^{k,\alpha }$ norm on the compact manifold $\mathcal {M}$ (for any $k \in \mathbb {N}_{0}$ and for any $\alpha \in (0,1)$) as follows (suppressing the dependence on $\mathscr {C}$):

$$\begin{aligned} \Vert u\Vert _{C^{k,\alpha }} := \sum _{i = 0}^{k} \sup _{\mathcal {M}} \Vert \nabla ^i u\Vert _g + \sum _{i=1}^N [\psi _i \nabla ^k u]_{\alpha ; B_i}. \end{aligned}$$

We denote the space of continuous sections of E for which this norm is finite by $C^{k,\alpha }(E)$.

Since the space of smooth tensor fields is not closed in the Hölder norms, in order to obtain the cleanest possible statements of our results, we also introduce the space of little Hölder continuous tensor fields. Let $h^{k,\alpha }(E)$ denote the completion of the set of smooth tensor fields with respect to the $C^{k,\alpha }$ norm. We note that the space $h^{k,\alpha }(E)$ is a proper closed subset of $C^{k,\alpha }(E)$ [28].

We now discuss differential operators, first on functions and then on tensor fields. Consider a linear differential operator P of order k acting on functions, given in local coordinates by

$$\begin{aligned} P u = \sum _{|I|=0}^k a_I(x) \partial ^I u, \end{aligned}$$

where I is a multi-index and $a_I$ is a sufficiently smooth coefficient function (the precise regularity is expounded below). The principal symbol of P is defined by

$$\begin{aligned} \sigma _{\xi }(P;x) := i^k \sum _{|I| = k} a_I(x) \xi ^{I}, \end{aligned}$$

where $\xi$ is a real vector. The operator P is defined to be elliptic if $\sigma _{\xi }(P;x) \ne 0$ for all x and $\xi \ne 0$. If P is a linear differential operator of order k acting on sections of a tensor bundle E, then in terms of local coordinate-basis tensor components (with indices such as “$\ell$” representing collective tensor indices, and with summation over repeated indices presumed) one has

$$\begin{aligned} (P u)_j = \sum _{|I|=0}^k {a^{\ell } _j}_I(x) \partial ^I u_{\ell }, \end{aligned}$$

with the corresponding (now tensorial) symbol

$$\begin{aligned} \sigma _{\xi }(P;x)^{\ell }_j = i^k \sum _{|I| = k} (a_j^\ell )_I(x) \xi ^{I}. \end{aligned}$$

P is elliptic if for all x and $\xi \ne 0$, $\sigma _{\xi }(P;x)^\ell _j$ is an isomorphism. P is strongly elliptic if there exists a constant $C > 0$ such that for all x, $\xi$ with $|\xi |=1$, and $\eta \in \mathbb C$,

$$\begin{aligned} \mathrm {Re} \; \sigma _{\xi }(P;x)_j^{\ell } \eta ^j \overline{\eta }_{\ell } \ge C |\eta |^2. \; \end{aligned}$$

If P has smooth coefficients, we may regard it as a map $P:C^{\infty }(E) \rightarrow C^{\infty }(E)$. The $L^2$-pairing for sections of E may then be used to define the (formal) adjoint $P^*$ of P. Specifically, $P^*$ is the differential operator of order k such that

$$\begin{aligned} (Pu,v) = (u, P^* v), \; \; \forall u, v \in C^{\infty }(E). \end{aligned}$$

By definition, P is formally self-adjoint if $P^* = P$.

We use the following basic results from elliptic theory. These can be found, for example, in Appendix A of [7].

Theorem 8

Let $(\mathcal {M},g)$ be a compact manifold and E a tensor bundle over $\mathcal {M}$. Let $P: C^{\infty }(E) \longrightarrow C^{\infty }(E)$ be a linear formally self-adjoint strongly elliptic geometric operator. Then the extension of P to an unbounded operator

$$\begin{aligned} P: L^2(E) \longrightarrow L^2(E) \end{aligned}$$

has a discrete $L^2$-spectrum consisting of a sequence of real-valued eigenvalues that diverge to $-\infty$.

In addition to this extension of the linear differential operator P to $L^2(E)$, it also extends naturally to an operator mapping $C^{k+l,\alpha }(E)$ to $C^{l,\alpha }(E)$, and mapping $L^{k+l,p}(E)$ to $L^{l,p}(E)$. For P acting on these Hölder and Sobolev spaces, one has the following elliptic estimates (see Theorem 40 in the Appendix of [7]):

Theorem 9

(Appendix Theorem 27, [7]) For the linear differential operator P described above, there exist positive constants $c_1$ and $c_2$ such that

• (1) for every $u \in C^{k+l,\alpha }(E)$,

  $$\begin{aligned} \Vert u \Vert _{C^{k+l,\alpha }} \le c_1( \Vert Pu \Vert _{C^{l,\alpha }} + \Vert u \Vert _{C^{0,\alpha }} ); \end{aligned}$$

• (2) for every $u \in L^{k+l,p}(E), \; 1< p < \infty$,

  $$\begin{aligned} \Vert u \Vert _{L^{k+l,p}} \le c_2( \Vert Pu \Vert _{L^{l,p}} + \Vert u \Vert _{L^p} ). \end{aligned}$$

Although it is not stated explicitly in the reference above, the constants $c_1$ and $c_2$ appearing in the above estimates depend on upper and lower bounds for the coefficients of the operator; see [17].