On an inverse curvature flow in two-dimensional space forms

TL;DR

This paper investigates a specific inverse curvature flow for convex curves in two-dimensional space forms, proving long-term existence, exponential convergence to circles, and deriving geometric inequalities and counterexamples.

Contribution

It introduces a new inverse curvature flow in 2D space forms, proves its long-term behavior, and applies it to geometric inequalities and conjecture counterexamples.

Findings

Solutions exist for all time and converge exponentially to a geodesic circle.

The flow leads to proofs of isoperimetric and weighted geometric inequalities.

Provides a counterexample to a conjecture of Girão-Pinheiro in the 2D case.

Abstract

We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and converge exponentially fast in the smooth topology to a standard round geodesic circle. This has a number of consequences: first, to prove the isoperimetricinequality; second, to establish a range of weighted geometric inequalities; and third, to give a counterexample to the $n=2$ case of a conjecture of Gir\~ao-Pinheiro.