On volume-preserving crystalline mean curvature flow

TL;DR

This paper investigates the global existence and regularity of volume-preserving crystalline curvature flow, especially in non-convex and non-smooth settings, highlighting the preservation of geometric symmetries related to the Wulff shape.

Contribution

It establishes global existence and regularity results for volume-preserving crystalline curvature flow in non-convex and non-smooth cases, leveraging symmetry properties of the Wulff shape.

Findings

Preservation of reflection symmetries in the flow.

Global existence for smooth anisotropies.

Global existence results for certain non-smooth anisotropies.

Abstract

In this work we consider the global existence of volume-preserving crystalline curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address global existence and regularity of the flow for smooth anisotropies. For the non-smooth case we establish global existence results for the types of anisotropies known to be globally well-posed.