The geometric triharmonic heat flow of immersed surfaces near spheres

作者：McCoy James; Parkins Scott; Wheeler Glen^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2017, 161: 44-86.

DOI：10.1016/j.na.2017.05.016

• Curvature flow
• Higher-order parabolic PDE

We consider closed immersed surfaces in 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 (3)root 3V(0)/4 pi, where V-0 denotes the signed enclosed volume of the initial data.

• 出版日期2017-9