The geometric triharmonic heat flow of immersed surfaces near spheres. (September 2017) - McCoy, James
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 ε -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 3 V 0 ∕ 4 π 3, where V 0 denotes the signed enclosed volume of the initial data.