Non-preserved curvature conditions under constrained mean curvature flows

TL;DR

This paper demonstrates through explicit examples that certain curvature conditions, like mean convexity and scalar curvature positivity, are not preserved under constrained mean curvature flows, challenging previous assumptions and folklore conjectures.

Contribution

It provides the first explicit counterexamples showing non-preservation of key curvature conditions under constrained mean curvature flows.

Findings

Mean convexity is not preserved under volume- or area-preserving mean curvature flows.

Positivity of scalar curvature is not preserved in these flows.

Traditional singularity analysis methods are ineffective for constrained mean curvature flows.

Abstract

We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean curvature flow.