Generalized motion of level sets by functions of their curvatures on Riemannian manifolds

TL;DR

This paper develops a framework for the evolution of level sets governed by curvature functions on Riemannian manifolds, establishing existence, uniqueness, and consistency of solutions under various curvature conditions.

Contribution

It introduces a generalized approach to level set evolution on Riemannian manifolds, extending classical curvature-driven flows to singular functions and non-Euclidean geometries.

Findings

Existence and uniqueness of viscosity solutions on nonnegatively curved manifolds.

Framework applies to mean curvature and Gaussian curvature evolutions.

Counterexamples show properties differ on negatively curved manifolds.

Abstract

We consider the generalized evolution of compact level sets by functions of their normal vectors and second fundamental forms on a Riemannian manifold M. The level sets of a function $u:M\to\mathbb{R}$ evolve in such a way whenever u solves an equation $u_{t}+F(Du, D^{2}u)=0$, for some real function F satisfying a geometric condition. We show existence and uniqueness of viscosity solutions to this equation under the assumptions that M has nonnegative curvature, F is continuous off Du=0, (degenerate) elliptic, and locally invariant by parallel translation. We then prove that this approach is geometrically consistent, hence it allows to define a generalized evolution of level sets by very general, singular functions of their curvatures. For instance, these assumptions on F are satisfied when F is given by the evolutions of level sets by their mean curvature (even in arbitrary codimension) or by their positive Gaussian curvature. We also prove that the generalized evolution is consistent with the classical motion by the corresponding function of the curvature, whenever the latter exists. When M is not of nonnegative curvature, the same results hold if one additionally requires that F is uniformly continuous with respect to D^2 u. Finally we give some counterexamples showing that several well known properties of the evolutions in $\mathbb{R}^{n}$ are no longer true when M has negative sectional curvature.