Parabolic Classical Curvature Flows

TL;DR

This paper studies classical curvature flows of convex surfaces in 3D, analyzing their parabolicity, integrability, and geometric properties, introducing new tools like the Radii of Curvature diagram.

Contribution

It characterizes conditions for parabolicity, integrability, and boundedness in curvature flows, and introduces the Radii of Curvature diagram as a new analytical tool.

Findings

Flow decouples to highest order, simplifying analysis.

Zeroth order terms form a Hamiltonian system, indicating integrability.

Conditions for bounded geometric quantities are established.

Abstract

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of curvature is a second order system of partial differential equations which we show decouples to highest order. We determine the conditions for this system to be parabolic and investigate the lower order terms. The zeroth order terms are shown to form a Hamiltonian system, which is therefore completely integrable. We find conditions on parabolic curvature flows that ensure boundedness of various geometric quantities and investigate some examples, including powers of mean curvature flow, Gauss curvature flow and mean radius of curvature flow, as well as the non-homogeneous Bloore flow. As a new tool we introduce the Radii of Curvature diagram of a surface and its canonical hyperbolic metric. The relationship between these and the properties of Weingarten surfaces are also discussed.