Volume preserving centro-affine normal flows

TL;DR

This paper investigates the long-term evolution of volume-preserving p-flows in higher-dimensional space, proving convergence of symmetric solutions to the unit ball under certain conditions, extending previous affine normal flow techniques.

Contribution

It extends Andrews' method to analyze the volume-preserving p-flow, demonstrating convergence to the unit ball for symmetric solutions in a new setting.

Findings

Symmetric solutions converge to the unit ball in the $C^{ abla}$ topology.

The convergence is modulo the group of special linear transformations.

The results apply for $1 \,\leq p < \frac{n+1}{n-1}$.

Abstract

We study the long time behavior of the volume preserving $p$-flow in $\mathbb{R}^{n+1}$ for $1\leq p<\frac{n+1}{n-1}$. By extending Andrews' technique for the flow along the affine normal, we prove that every centrally symmetric solution to the volume preserving $p$-flow converges sequentially to the unit ball in the $C^{\infty}$ topology, modulo the group of special linear transformations.