SINGULARITIES AND FULL CONVERGENCE OF THE MÖBIUS-INVARIANT WILLMORE FLOW IN THE 3-SPHERE

Abstract

In this article we continue our investigation of the Möbius-invariant Willmore flow (MIWF), a particular variant of the ordinary Willmore flow which is equivariant with respect to Möbius-transformations of its ambient space-Rn respectively Sn. As a matter of fact, the flow lines of the MIWF have to start moving in umbilic-free immersions F0 which map some fixed compact torus Σ into the fixed ambient space of the considered flow. Here we investigate the behaviour of flow lines {Ft} of the MIWF in S3 starting with relatively low Willmore energy, as the time t approaches the maximal time of existence Tmax(F0) > 0 of {Ft}. On the one hand we detect divergent flow lines of the MIWF, and on the other hand we investigate the formation of limit surfaces of both divergent and convergent flow lines. Such a limit surface is the support of a certain integral 2varifold µ in R4, arising as a measure-theoretic limit of the sequence of varifolds {H2⌊Ftl (Σ)}, for an appropriately chosen sequence tl ↗ Tmax(F0), and it turns out to be homeomorphic to either a 2-sphere or a compact torus in S3, provided it is not the empty set. In the special case in which the limit surface spt(µ) is a compact torus, this surface can be parametrized by a uniformly conformal bi-Lipschitz homeomorphism (Formula Presented), and under certain additional conditions on the sequence {Ftl } such a uniformly conformal parametrization f turns out to be a diffeomorphism of class W4,2(Σ, R4). Finally, if the initial immersion F0 of a flow line {Ft} is assumed to parametrize a smooth Hopf-torus in S3 with Willmore energy not bigger than 8π, then we obtain more precise statements about the flow line {Ft} as t ↗ Tmax(F0), especially stronger types of convergence of particular subsequences of {Ft} to uniformly conformal W4,2-parametrizations of limit Hopf-tori. This insight will finally yield a new criterion for full smooth convergence of such flow lines of the MIWF to smooth diffeomorphisms parametrizing the Clifford torus, up to Möbiustransformations of S3.