2022/05/02 by Jakob, Ruben
#30C35 #35R01 #53C18 #53C42 #53E40 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2205.00604
Here we continue the investigation of the Möbius-invariant Willmore flow (MIWF), starting to move in arbitrary smooth and umbilic-free initial immersions F0 which map some fixed compact torus Σ into ℝn respectively \mathbbSn. Here we investigate the behaviour of flow lines \Ft\ of the MIWF in \mathbbS3 starting with relatively low Willmore energy, as the time t approaches the maximal time of existence Tmax(F0) of \Ft\. We succeed to construct divergent flow lines, and we investigate limit surfaces of both divergent and convergent flow lines of the MIWF. At least generically a limit surface of some general flow line \Ft\ of the MIWF can be identified with the support of an integral 2-varifold μ in ℝ4, which is the weak limit of the sequence of varifolds \H2\lfloor_F_tl(Σ)\, for an appropriately chosen sequence tl \nearrow Tmax(F0), and that spt(μ) is either empty or homeomorphic to some compact, closed manifold of genus either 0 or 1. In the particular case in which spt(μ) is a compact surface of genus 1 it can be parametrized by a uniformly conformal bi-Lipschitz homeomorphism f of class W2,2∩ W1,∞, and under certain additional conditions on \F_tl\ such a parametrization is a diffeomorphism of class W4,2. Finally, if the initial immersion F0 of a flow line \Ft\ is assumed to parametrize a Hopf-torus in \mathbbS3 with Willmore energy not bigger than 8 π, then we obtain more precise statements about the flow line \Ft\ as t \nearrow Tmax(F0). This insight will finally yield a criterion for full convergence of such flow lines of the MIWF to parametrizations of the Clifford torus - up to Möbius-transformations of \mathbbS3 - as t \nearrow ∞.