A fully-nonlinear flow and quermassintegral inequalities in the sphere

TL;DR

This paper reviews fully nonlinear curvature flows in spherical space that preserve certain geometric quantities, aiming to prove inequalities related to quermassintegrals, with some convergence results known but a full proof still pending.

Contribution

It compiles current knowledge on locally constrained curvature flows in the sphere and highlights open problems related to quermassintegral inequalities.

Findings

Flow convergence to a round sphere is conjectured but not yet proven.

Flow preserves one quermassintegral while deforming others.

The paper consolidates known results and emphasizes open problems.

Abstract

This expository paper presents the current knowledge of particular fully nonlinear curvature flows with local forcing term, so-called locally constrained curvature flows. We focus on the spherical ambient space. The flows are designed to preserve a quermassintegral and to de-/increase the other quermassintegrals. The convergence of this flow to a round sphere would settle the full set of quermassintegral inequalities for convex domains of the sphere, but a full proof is still missing. Here we collect what is known and hope to attract wide attention to this interesting problem.