[Paper Review] Surfaces moving by powers of Gauss curvature

Ben Andrews, Xuzhong Chen|arXiv (Cornell University)|Nov 20, 2011

Geometric Analysis and Curvature Flows11 references4 citations

TL;DR

This paper establishes that strictly convex surfaces evolving under the flow $ X_t = -K^{eta/2} \nu $ with $ \beta \in [1,2] $ converge to a sphere upon shrinking to a point. The authors introduce a novel curvature pinching quantity derived from solving the reaction ODE system of curvature evolution, which is preserved under the flow and enables a sharp pinching estimate that ensures spherical limit shape.

ABSTRACT

We prove that strictly convex surfaces moving by $K^{α/2}$ become spherical as they contract to points, provided $α$ lies in the range $[1,2]$. In the process we provide a natural candidate for a curvature pinching quantity for surfaces moving by arbitrary functions of curvature, by finding a quantity conserved by the reaction terms in the evolution of curvature.

Motivation & Objective

• To prove that surfaces evolving by $ K^{eta/2} $ for $ \beta \in [1,2] $ converge smoothly to a round sphere upon shrinking to a point.
• To develop a general method for identifying curvature pinching quantities conserved under arbitrary curvature-driven flows.
• To extend the known pinching estimates for curvature evolution beyond the known cases of Gauss curvature flow and mean curvature flow.
• To provide a systematic approach to curvature pinching that applies to flows of arbitrary homogeneity degree, particularly for powers of Gauss curvature.
• To demonstrate that the proposed pinching quantity is effective in proving spherical limit behavior for flows with $ \beta \in [1,2] $

Proposed method

• Uses the support function representation of convex surfaces to derive a scalar parabolic PDE for the evolution of $ s(z,t) $, the support function.
• Derives the evolution equation for the Weingarten map $ \mathfrak{r}_{ij} $, which governs principal curvatures, using the Gauss-Weingarten structure equations.
• Identifies a conserved quantity $ F(\mathfrak{r}_2 - \mathfrak{r}_1) = c $, where $ F = -\det(\mathfrak{r})^{-\alpha/2} $, by solving the reaction ODE system of curvature evolution.
• Applies Hamilton’s maximum principle to show that the pinching quantity $ \kappa_2 - \kappa_1 $ remains bounded relative to curvature, ensuring spherical convergence.
• Computes the gradient and zero-order terms in the evolution of the pinching quantity and proves their non-positivity for $ \alpha \in [1,2] $, ensuring monotonicity.
• Uses the resulting pinching estimate to deduce that the rescaled flow converges smoothly to a sphere, leveraging known regularity and convergence results from prior work.

Experimental results

Research questions

• RQ1 Can a curvature pinching quantity be systematically derived for general curvature-driven flows that ensures spherical limit shapes?
• RQ2 Does the flow $ X_t = -K^{\alpha/2} \nu $ with $ \alpha \in [1,2] $ lead to spherical convergence for strictly convex initial surfaces?
• RQ3 What conditions on the speed function $ F $ ensure that the gradient terms in the curvature evolution preserve a pinching quantity?
• RQ4 For which values of $ \alpha $ does the pinching estimate fail, and what does this imply about the limiting shape?
• RQ5 Can the proposed method be extended to other flows such as those by powers of mean curvature or norm of the second fundamental form?

Key findings

• For $ \alpha \in [1,2] $, any strictly convex surface evolving by $ X_t = -K^{\alpha/2} \nu $ converges smoothly to a point and, upon rescaling, converges in $ C^\infty $ to a unit sphere.
• The authors construct a natural curvature pinching quantity $ F(\mathfrak{r}_2 - \mathfrak{r}_1) = c $, which is conserved under the flow and serves as a key tool in proving the pinching estimate.
• The pinching estimate fails for $ \alpha > 2 $ when the ratio of principal curvatures becomes large, indicating that spherical convergence does not hold in general beyond this range.
• For $ \alpha = 1/2 $, the method yields a valid pinching estimate, and the flow corresponds to the affine-invariant flow, which is known to preserve convexity and converge to a limit shape.
• The method extends to other flows: for $ H^\alpha $, the quantity $ H^\alpha |\kappa_2 - \kappa_1| / K $ is non-increasing for $ \alpha \leq \alpha_* \approx 5.16 $; for $ |A|^\alpha $, the estimate holds up to $ \alpha \approx 8.15 $.
• For $ F = \kappa_1^\alpha + \kappa_2^\alpha $ with $ \alpha > 1 $, the pinching estimate $ F |\kappa_2 - \kappa_1| / K \leq C $ holds, marking the first such result for flows of arbitrarily high homogeneity degree.