The geometric triharmonic heat flow of immersed surfaces near spheres. (September 2017)

• Record Type: Journal Article
• Authors: McCoy, James; Parkins, Scott; Wheeler, Glen
• Is Part Of: Nonlinear analysis. Volume 161(2017)
• Page Start: 44
• Page End: 86
• Publication Date: 2017-09
• Subjects: Curvature flow -- Higher-order parabolic PDE; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2017.05.016
• Languages: English
• ISSNs: 0362-546X

Abstract

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.