[Paper Review] A mean curvature type flow in space forms

Pengfei Guan, Junfang Li|arXiv (Cornell University)|Sep 19, 2013

Geometric Analysis and Curvature Flows11 references4 citations

TL;DR

This paper introduces a novel mean curvature-type flow in space forms that evolves star-shaped hypersurfaces into round spheres without curvature assumptions, preserving enclosed volume and monotonically decreasing surface area. The flow ensures exponential convergence in $C^rown$ topology and establishes monotonicity of quermassintegrals, leading to a new proof of Alexandrov-Fenchel inequalities for convex domains in $\mathbb{R}^{n+1}$.

ABSTRACT

In this article, we introduce a new type of mean curvature flow for bounded star-shaped domains in space forms and prove its longtime existence, exponential convergence without any curvature assumption. Along this flow, the enclosed volume is a constant and the surface area evolves monotonically. Moreover, for a bounded convex domain in R n+1, the quermassintegrals evolve monotonically along the flow which allows us to prove a class of Alexandrov-Fenchel inequalities of quermassintegrals.

Motivation & Objective

• To develop a new geometric flow in space forms that evolves star-shaped hypersurfaces into round spheres without curvature assumptions.
• To prove longtime existence and exponential convergence of the flow in $C^\infty$ topology.
• To establish monotonicity of quermassintegrals along the flow for convex domains in $\mathbb{R}^{n+1}$.
• To derive a new proof of the classical Alexandrov-Fenchel inequalities using the flow's monotonicity properties.
• To generalize existing results on geometric inequalities via a flow-based approach, avoiding reliance on curvature pinching or convexity beyond starlikeness.

Proposed method

• The flow is defined by the evolution equation $\partial_t X = (n\phi'(\rho) - Hu)\nu$ in space forms, where $H$ is mean curvature, $u = \langle X, \nu \rangle$ is the support function, and $\phi$ encodes the space form geometry.
• The flow preserves the enclosed volume and monotonically decreases surface area, linking it to the isoperimetric problem.
• The Minkowski identity is used as a key tool to derive the flow's structure and ensure parabolicity under star-shapedness.
• A priori estimates are established using a maximum principle argument on the support function and Weingarten curvature, with careful decomposition of the Hessian into low- and high-eigenvalue regions.
• The proof of long-time existence relies on a modified maximum principle for the quantity $\phi = \text{dist}(x, \partial M_t)$, controlling curvature growth.
• The monotonicity of quermassintegrals is derived from the evolution of elementary symmetric functions of principal curvatures under the flow.

Experimental results

Research questions

• RQ1 Can a mean curvature-type flow be constructed in space forms that evolves star-shaped hypersurfaces into spheres without curvature assumptions?
• RQ2 Does the flow preserve volume and monotonically decrease surface area, thereby linking to the isoperimetric problem?
• RQ3 Can the monotonicity of quermassintegrals along the flow be used to prove Alexandrov-Fenchel inequalities for convex domains in $\mathbb{R}^{n+1}$?
• RQ4 How does the new flow compare to normalized or volume-preserving mean curvature flows in terms of curvature assumptions and long-term behavior?
• RQ5 What role does the Minkowski identity play in constructing and analyzing the flow’s behavior?

Key findings

• The flow $\partial_t X = (n\phi'(\rho) - Hu)\nu$ exists for all time $t \in [0, \infty)$ and converges exponentially to a round sphere in $C^\infty$ topology for any smooth, star-shaped initial hypersurface in space forms.
• The enclosed volume remains constant along the flow, while surface area decreases monotonically.
• For convex domains in $\mathbb{R}^{n+1}$, all quermassintegrals $\int_{M} \sigma_k(\kappa) \, d\mu_g$ are non-increasing along the flow.
• The monotonicity of quermassintegrals leads to a new proof of the Alexandrov-Fenchel inequalities: $V(\Omega)^{1/(n+1)} \leq c_{n,k} \left( \int_{\partial\Omega} \sigma_k(\kappa) \, d\mu \right)^{1/(n-k)}$ for $0 \leq k < n-1$, with equality iff $\Omega$ is a ball.
• The flow preserves convexity and becomes strictly convex instantly for convex initial data, even without initial strict convexity.
• The method avoids curvature pinching and relies only on starlikeness, making it applicable to a broader class of initial hypersurfaces than previous flows.