The geometric triharmonic heat flow of immersed surfaces near spheres

TL;DR

This paper studies the evolution of immersed surfaces in three-dimensional space under the geometric triharmonic heat flow, establishing regularity, singularity behavior, and convergence to spheres based on initial curvature conditions.

Contribution

It introduces new interior estimates, a gap lemma for stationary solutions, and proves exponential convergence to spheres for flows with small initial curvature.

Findings

Flow solutions have a positive lifespan depending on initial curvature.

Blowup analysis shows asymptotic approach to non-umbilic stationary surfaces.

Solutions with small initial L^2-norm of tracefree curvature converge exponentially to spheres.

Abstract

We consider closed immersed surfaces in R^3 evolving by the geometric triharmonic heat flow. Using local energy estimates, we prove interior estimates and a positive absolute lower bound on the lifespan of solutions depending solely on the local concentration of curvature of the initial immersion in L^2. We further use an {\epsilon}-regularity type result to prove a gap lemma for stationary solutions. Using a monotonicity argument, we then prove that a blowup of the flow approaching a singular time is asymptotic to a non-umbilic embedded stationary surface. This allows us to conclude that any solution with initial L^2-norm of the tracefree curvature tensor smaller than an absolute positive constant converges exponentially fast to a round sphere with radius equal to the cube root of 3V_0/4{\pi}, where V_0 denotes the signed enclosed volume of the initial data.