Arnold-Thom gradient conjecture for the arrival time

2017/12/14 by Tobias Holck Colding, Colding, Tobias Holck, William P. Minicozzi II +1 · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AG #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.1712.05381

arxiv created 2017/12/14 · arxiv updated 2017/12/15

Abstract

We prove conjectures of Rene Thom and Vladimir Arnold for C² 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.

Cited by

• Non-uniqueness of geodesic limits and a question of Grayson and Gage

Related

• Arnold-Thom conjecture for the arrival time of surfaces
• Thom's gradient conjecture for parabolic systems and the Yang-Mills and Ricci flows
• Level set method for motion by mean curvature
• Thom's gradient conjecture for nonlinear evolution equations
• Motion Of Level Set By General Curvature