Global existence and full convergence of the M\"obius-invariant Willmore flow in the $3$-sphere

Abstract

In this article, we prove two "global existence and full convergence theorems" for flow lines of the M\"obius-invariant Willmore flow, and we use these results, in order to prove that fully and smoothly convergent flow lines of the M\"obius-invariant Willmore flow are stable w.r.t. small perturbations of their initial immersions in any $C^{4,\gamma}$-norm, provided they converge either to a smooth parametrization of "a Clifford-torus" in $\mathbb{S}^3$ or to a umbilic-free $C^4$-local minimizer of the Willmore functional in either $\mathbb{R}^3$ or $\mathbb{S}^3$. The proofs of our four main theorems rely on the author's recent achievements about the M\"obius-invariant Willmore flow, on Escher's, Mayer's and Simonett's work from "the 90s" on "invariant center manifolds" for uniformly parabolic quasilinear evolution equations and their special applications to the "Willmore flow" and "Surface diffusion flow" near round $2$-spheres in $\mathbb{R}^3$ and on Riviere's and Bernard's fundamental investigation of the Willmore functional on the basis of its conformal invariance and Noether's Theorem.