Avoidance for Set-Theoretic Solutions of Mean-Curvature-Type Flows

TL;DR

This paper extends the theory of set-theoretic solutions to mean-curvature-type flows to general Riemannian manifolds, proving disjointness preservation under broader conditions than previously known.

Contribution

It generalizes the disjointness preservation property for set-theoretic subsolutions of mean curvature flows to arbitrary Riemannian manifolds with ambient vector fields.

Findings

Disjointness is preserved for set-theoretic subsolutions in general Riemannian manifolds.

The result extends previous Euclidean-only theorems to more general geometric settings.

The paper corrects a minor typo in the latest version.

Abstract

We provide a self-contained treatment of set-theoretic subsolutions to flow by mean curvature, or, more generally, to flow by mean curvature plus an ambient vector field. The ambient space can be any smooth Riemannian manifold. Most importantly, we show that if two such set-theoretic subsolutions are initially disjoint, then they remain disjoint as long as one of the subsolutions is compact; previously, this was only known for Euclidean space (with no ambient vectorfield). The new version (Sept 15, 2023) corrects a minor typo.