Highly Degenerate Harmonic Mean Curvature Flow

2008/04/24 by Michèle Caputo, Panagiota Daskalopoulos, Caputo, M. Cristina +1
Engineering · Mathematics · #Aerodynamics and Fluid Dynamics Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.0804.3936

openalex publication_date 2008/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the evolution of a weakly convex surface Σ₀ in \R³ with flat sides by the Harmonic Mean Curvature flow. We establish the short time existence as well as the optimal regularity of the surface and we show that the boundaries of the flat sides evolve by the curve shortening flow. It follows from our results that a weakly convex surface with flat sides of class C^k,γ, for some k∈ ℕ and 0 < γ≤ 1, remains in the same class under the flow. This distinguishes this flow from other, previously studied, degenerate parabolic equations, including the porous medium equation and the Gauss curvature flow with flat sides, where the regularity of the solution for t >0 does not depend on the regularity of the initial data.

Related

• The harmonic mean curvature flow of nonconvex surfaces in ℝ³
• Mean curvature flow with generic initial data
• General curvature flow without singularities
• Lectures on mean curvature flow of surfaces
• Inverse Mean Curvature Flow with Singularities