A fully nonlinear flow for two-convex hypersurfaces in Riemannian manifolds

Abstract

We consider a one-parameter family of closed, embedded hypersurfaces moving with normal velocity $G_\kappa = \big ( \sum _{i < j} \frac{1}{\lambda _i+\lambda _j-2\kappa } \big )^{-1}$, where $\lambda _1 \le \cdots \le \lambda _n$ denote the curvature eigenvalues and $\kappa$ is a nonnegative constant. This defines a fully nonlinear parabolic equation, provided that $\lambda _1+\lambda _2>2\kappa$. In contrast to mean curvature flow, this flow preserves the condition $\lambda _1+\lambda _2>2\kappa$ in a general ambient manifold. Our main goal in this paper is to extend the surgery algorithm of Huisken–Sinestrari to this fully nonlinear flow. This is the first construction of this kind for a fully nonlinear flow. As a corollary, we show that a compact Riemannian manifold satisfying $\overline{R}_{1313}+\overline{R}_{2323} \ge -2\kappa ^2$ with non-empty boundary satisfying $\lambda _1+\lambda _2 > 2\kappa$ is diffeomorphic to a 1-handlebody. The main technical advance is the pointwise curvature derivative estimate. The proof of this estimate requires a new argument, as the existing techniques for mean curvature flow due to Huisken–Sinestrari, Haslhofer–Kleiner, and Brian White cannot be generalized to the fully nonlinear setting. To establish this estimate, we employ an induction-on-scales argument; this relies on a combination of several ingredients, including the almost convexity estimate, the inscribed radius estimate, as well as a regularity result for radial graphs. We expect that this technique will be useful in other situations as well.

Appendix: Review of Krylov–Safonov estimates

For the convenience of the reader, we collect some well-known regularity results for parabolic equations. The first one is the crucial Hölder estimate of Krylov and Safonov [30] (see also [29], Theorem 7 on pp. 137–138):

Theorem A.1

(N.V. Krylov, M.V. Safonov) Let $v: B_1(0) \times [0,1] \rightarrow {\mathbb {R}}$ be a solution of the parabolic equation $\frac{\partial }{\partial t} v = \sum _{i,j} a_{ij} \, D_i D_j v + \sum _i b_i \, D_i v + f$. We assume that the coefficients satisfy $\frac{1}{K} \, \delta _{ij} \le a_{ij} \le K \, \delta _{ij}$ and $|b_i| \le K$. Then

$$\begin{aligned}{}[v]_{C^{\gamma ;\frac{\gamma }{2}}(B_{\frac{1}{2}}(0) \times [\frac{1}{2},1])}\le & {} C \, \left( \sup _{B_1(0) \times [0,1])} v - \inf _{B_1(0) \times [0,1]} v \right) \\&\quad + C \, \Vert f\Vert _{C^0(B_1(0) \times [0,1])}, \end{aligned}$$

where $\gamma >0$ and $C>0$ depend only on K.

Corollary A.2

(N.V. Krylov, M.V. Safonov) Let $0 < \tau \le \frac{1}{4}$, and let $v: B_1(0) \times [0,\tau ] \rightarrow {\mathbb {R}}$ be a solution of the parabolic equation $\frac{\partial }{\partial t} v = \sum _{i,j} a_{ij} \, D_i D_j v + \sum _i b_i \, D_i v + f$. We assume that the coefficients satisfy $\frac{1}{K} \, \delta _{ij} \le a_{ij} \le K \, \delta _{ij}$ and $|b_i| \le K$. Finally, we assume that $\Vert v\Vert _{C^0(B_1(0) \times [0,\tau ])} + \Vert v(\cdot ,0)\Vert _{C^2(B_1(0))} + \Vert f\Vert _{C^0(B_1(0) \times [0,\tau ])} \le L$. Then

$$\begin{aligned}{}[v]_{C^{\gamma ;\frac{\gamma }{2}}(B_{\frac{1}{2}}(0) \times [0,\tau ])} \le C. \end{aligned}$$

Here, $\gamma >0$ depends only on K, and C depends only on K and L. In particular, $\gamma$ and C are independent of $\tau$.

Proof

We sketch the argument for the convenience of the reader. Using a straightforward barrier argument, we can show that

$$\begin{aligned} \sup _{B_r(x) \times [0,\min \{r^2,\tau \}]} v \le v(x,0)+Cr \end{aligned}$$

and

$$\begin{aligned} \inf _{B_r(x) \times [0,\min \{r^2,\tau \}]} v \ge v(x,0)-Cr \end{aligned}$$

for $x \in B_{\frac{1}{2}}(0)$ and $0 < r \le \frac{1}{2}$. This gives

$$\begin{aligned} \sup _{B_r(x) \times [0,\min \{r^2,\tau \}]} v - \inf _{B_r(x) \times [0,\min \{r^2,\tau \}]} v \le Cr \end{aligned}$$

(6)

for $x \in B_{\frac{1}{2}}(0)$ and $0 < r \le \frac{1}{2}$. Using Theorem A.1 together with (6), we obtain

$$\begin{aligned}{}[v]_{C^{\gamma ;\frac{\gamma }{2}}(B_{\frac{r}{2}}(x) \times [\frac{r^2}{2},r^2])} \le C \, r^{1-\gamma } \end{aligned}$$

(7)

for $x \in B_{\frac{1}{2}}(0)$ and $0 < r \le \tau ^{\frac{1}{2}}$. We now consider two points (x, t) and $(\tilde{x},\tilde{t})$ in spacetime such that $t \ge \tilde{t} \ge 0$. If $2 \, |x-\tilde{x}| + 2 \, (t-\tilde{t})^{\frac{1}{2}} < t^{\frac{1}{2}}$, then (7) gives

$$\begin{aligned} |v(x,t) - v(\tilde{x},\tilde{t})| \le C \, (|x-\tilde{x}| + (t-\tilde{t})^{\frac{1}{2}})^\gamma . \end{aligned}$$

On the other hand, if $2 \, |x-\tilde{x}| + 2 \, (t-\tilde{t})^{\frac{1}{2}} \ge t^{\frac{1}{2}}$, then (6) implies

$$\begin{aligned} |v(x,t) - v(\tilde{x},\tilde{t})| \le C \, (|x-\tilde{x}| + t^{\frac{1}{2}}) \le C \, (3 \, |x-\tilde{x}| + 2 \, (t-\tilde{t})^{\frac{1}{2}}). \end{aligned}$$

Putting these facts together, the assertion follows.

Combining the Krylov–Safonov estimate with the deep work of Evans [16], Krylov [28], and Caffarelli [13] on fully nonlinear elliptic equations gives:

Theorem A.3

Let $u: B_1(0) \times [0,1] \rightarrow {\mathbb {R}}$ be a solution of a fully nonlinear parabolic equation

$$\begin{aligned} \frac{\partial }{\partial t} u = \Phi (D^2 u,Du,u,x), \end{aligned}$$

where $\Phi$ depends smoothly on all its arguments. We assume that u is bounded in $C^{2;1}(B_1(0) \times [0,1])$. Moreover, we assume that the equation is uniformly parabolic, and $\Phi$ is concave in the first argument. Then u is uniformly bounded in $C^{2,\gamma ;1,\frac{\gamma }{2}}(B_{\frac{1}{4}}(0) \times [\frac{1}{2},1])$ for some uniform constant $\gamma >0$.

Proof

Consider the function $v = \frac{\partial }{\partial t} u$. The function v satisfies a uniformly parabolic equation. Moreover, v is bounded in $C^0(B_1(0) \times [0,1])$, so the Krylov–Safonov estimate implies that v is bounded in $C^{\gamma ;\frac{\gamma }{2}}(B_{\frac{1}{2}}(0) \times [\frac{1}{2},1])$. Using Theorem 3 in [13], it follows that $\sup _{t \in [\frac{1}{2},1]} \Vert u(t)\Vert _{C^{2,\gamma }(B_{\frac{1}{4}}(0))}$ is bounded from above. In other words, $D^2 u$ is uniformly Hölder continuous in space. Finally,

$$\begin{aligned}&\Vert D^2 u(t)-D^2 u(t')\Vert _{C^0(B_{\frac{1}{4}}(0))} \le C \, \Vert D^2 u(t)\\&\quad -D^2 u(t')\Vert _{C^\gamma (B_{\frac{1}{4}}(0))}^{\frac{2}{2+\gamma }} \, \Vert u(t)-u(t')\Vert _{C^0(B_{\frac{1}{4}}(0))}^{\frac{\gamma }{2+\gamma }} \le C \, |t-t'|^{\frac{\gamma }{2+\gamma }} \end{aligned}$$

for $t,t' \in [\frac{1}{2},1]$. This shows that $D^2 u$ is uniformly Hölder continuous in time.

Corollary A.4

Let $0 < \tau \le \frac{1}{4}$, and let $u: B_1(0) \times [0,\tau ] \rightarrow {\mathbb {R}}$ be a solution of a fully nonlinear parabolic equation

$$\begin{aligned} \frac{\partial }{\partial t} u = \Phi (D^2 u,Du,u,x), \end{aligned}$$

where $\Phi$ depends smoothly on all its arguments. We assume that u is bounded in $C^{2;1}(B_1(0) \times [0,\tau ])$, and that the initial function $u(\cdot ,0)$ is bounded in $C^4(B_1(0))$. Moreover, we assume that the equation is uniformly parabolic, and $\Phi$ is concave in the first argument. Then u is uniformly bounded in $C^{2,\gamma ;1,\frac{\gamma }{2}}(B_{\frac{1}{4}}(0) \times [0,1])$ for some uniform constant $\gamma >0$.

Proof

We again consider the function $v = \frac{\partial }{\partial t} u$. The function v satisfies a uniformly parabolic equation. Moreover, v is bounded in $C^0(B_1(0) \times [0,1])$ and the initial function $v(\cdot ,0)$ is bounded in $C^2(B_1(0))$. Consequently, v is bounded in $C^{\gamma ;\frac{\gamma }{2}}(B_{\frac{1}{2}}(0) \times [0,1])$. As above, Theorem 3 in [13] implies that $\sup _{t \in [0,1]} \Vert u(t)\Vert _{C^{2,\gamma }(B_{\frac{1}{4}}(0))}$ is uniformly bounded from above. In other words, $D^2 u$ is uniformly Hölder continuous in space. As above, we have the estimate

$$\begin{aligned}&\Vert D^2 u(t)-D^2 u(t')\Vert _{C^0(B_{\frac{1}{4}}(0))} \le C \, \Vert D^2 u(t)\\&\quad -D^2 u(t')\Vert _{C^\gamma (B_{\frac{1}{4}}(0))}^{\frac{2}{2+\gamma }} \, \Vert u(t)-u(t')\Vert _{C^0(B_{\frac{1}{4}}(0))}^{\frac{\gamma }{2+\gamma }} \le C \, |t-t'|^{\frac{\gamma }{2+\gamma }} \end{aligned}$$

for $t,t' \in [0,1]$. Hence, $D^2 u$ is uniformly Hölder continuous in time, as claimed.