Arnold-Thom conjecture for the arrival time of surfaces

TL;DR

This paper proves Arnold's conjecture and Łojasiewicz's theorem regarding the limit tangents of gradient flow lines for arrival time functions in mean curvature flows, including cases with singularities and non-smooth functions.

Contribution

It establishes the conjectures for mean curvature flows with singularities, extending previous results to non-$C^2$ arrival time functions and specific initial surfaces.

Findings

Proves Łojasiewicz's theorem for arrival time functions.

Validates Arnold's conjecture for flows with neck or cylindrical singularities.

Applies results to flows starting from spheres and generic surfaces.

Abstract

Following \L ojasiewicz's uniqueness theorem and Thom's gradient conjecture, Arnold proposed a stronger version about the existence of limit tangents of gradient flow lines for analytic functions. We prove \L ojasiewicz's theorem and Arnold's conjecture in the context of arrival time functions for mean curvature flows in $\mathbb R^{n+1}$ with neck or non-degenerate cylindrical singularities. In particular, we prove the conjectures for all mean convex mean curvature flows of surfaces, including the cases when the arrival time functions are not $C^2.$ The results also apply to mean curvature flows starting from two-spheres or generic closed surfaces.