Asymptotic convergence for modified scalar curvature flow

TL;DR

This paper investigates a geometric flow of starshaped hypersurfaces driven by a modified scalar curvature term, proving long-term existence, preservation of shape, and exponential convergence to a sphere for certain parameters.

Contribution

It establishes asymptotic convergence results for a new class of scalar curvature flows with a specific speed function, extending previous understanding of geometric evolution.

Findings

Flow exists for all time and preserves starshapedness.

Normalized flow converges exponentially to a sphere for α ≥ 2.

Counterexample shows non-convergence when α < 2.

Abstract

In this paper, we study the flow of closed, starshaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^\alpha\sigma_2^{1/2},$ where $\sigma_2^{1/2}$ is the normalized square root of the scalar curvature, $\alpha\geq 2,$ and $r$ is the distance from points on the hypersurface to the origin. We prove that the flow exists for all time and the starshapedness is preserved. Moreover, after normalization, we show that the flow converges exponentially fast to a sphere centered at origin. When $\alpha<2,$ a counterexample is given for the above convergence.