Arnold-Thom gradient conjecture for the arrival time

TL;DR

This paper proves longstanding conjectures by Thom and Arnold for solutions to a degenerate elliptic equation related to mean curvature motion, suggesting such solutions behave analytically despite degeneracy.

Contribution

It establishes the Arnold-Thom conjecture for C^2 solutions to a key degenerate elliptic equation, revealing a potential general principle about degenerate equations.

Findings

Proves Arnold-Thom conjecture for mean curvature level set equation

Demonstrates solutions behave as if they are analytic

First results indicating a broader principle for degenerate equations

Abstract

We prove conjectures of Rene Thom and Vladimir Arnold for C^2 solutions to the degenerate elliptic equation that is the level set equation for motion by mean curvature. We believe these results are the first instances of a general principle: Solutions of many degenerate equations behave as if they are analytic, even when they are not. If so, this would explain various conjectured phenomena.