A unique continuation property for the level set equation

TL;DR

This paper proves a unique continuation property for the level set equation in mean curvature flow, showing that if a solution matches a ball's solution to infinite order at a maximum point, then the entire domain must be a ball.

Contribution

It establishes a new unique continuation result for the level set equation in mean curvature flow, characterizing when solutions coincide with spherical solutions.

Findings

Solutions matching a ball to infinite order at a maximum point imply the domain is a ball.

The result characterizes the geometric structure of solutions in mean curvature flow.

Unique continuation holds under the specified conditions, ensuring rigidity of solutions.

Abstract

We prove the following unique continuation result: if a solution to the level set equation for mean curvature flow in a mean-convex domain agrees to infinite order at the point where it attains its maximum with the solution for a ball, then it agrees everywhere and the domain is a ball.