When maximum-principle functions cease to exist

TL;DR

This paper investigates the existence of maximum-principle functions in geometric flow equations for convex surfaces, identifying conditions under which these functions do or do not exist, and introduces new such functions.

Contribution

The paper characterizes when maximum-principle functions exist or cease to exist in geometric flows and presents newly discovered functions with this property.

Findings

Identifies conditions for existence and non-existence of maximum-principle functions.

Introduces new maximum-principle functions for geometric flows.

Provides insights into the convergence of convex surfaces to spheres.

Abstract

We consider geometric flow equations for contracting and expanding normal velocities, including powers of the Gauss curvature, of the mean curvature, and of the norm of the second fundamental form, and ask whether - after appropriate rescaling - closed strictly convex surfaces converge to spheres. To prove this, many authors use certain functions of the principal curvatures, which we call maximum-principle functions. We show when such functions cease to exist and exist, while presenting newly discovered maximum-principle functions.