2020/02/03 by Ruben Jakob, Jakob, Ruben
Mathematics · #11Z05 #35R01 #53C42 #53E40 #58J35 #Analysis of PDEs (math.AP) #FOS: Mathematics #G.0 #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology
paper · doi:10.48550/arxiv.2002.01006
openalex publication_date 2020/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, the author investigates flow lines of the classical Willmore flow, which start to move in a smooth parametrization of a Hopf-torus in \mathbbS3. We prove that any such flow line of the Willmore flow exists globally, in particular does not develop any singularities, and subconverges to some smooth Willmore-Hopf-torus in every Cm-norm. Moreover, if in addition the Willmore energy of the initial immersion F0 is required to be smaller than or equal to the threshold (8π2)/(√(2)), then the unique flow line of the Willmore flow, starting to move in F0, converges fully to a conformally transformed Clifford torus in every Cm-norm, up to time dependent, smooth reparametrizations. Key instruments for the proofs are the equivariance of the Hopf-fibration π:\mathbbS3 \longrightarrow \mathbbS2 w.r.t. the effect of the L2-gradient of the Willmore energy applied to smooth Hopf-tori in \mathbbS3 and to smooth closed regular curves in \mathbbS2, a particular version of the Lojasiewicz-Simon gradient inequality, and a well-known classification and description of smooth, arc-length parametrized solutions of the Euler-Lagrange equation of the elastic energy functional in terms of Jacobi Elliptic Functions and Elliptic Integrals, dating back to the 80s.