The Willmore flow of Hopf-tori in the 33-sphere

In this article the author investigates flow lines of the classical Willmore flow, which start moving in a parametrization of a Hopf-torus in $\mathbb{S}^3$. 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 $C^{m}$-norm. Moreover, if in addition the Willmore-energy of the initial immersion $F_0$ is required to be smaller than the threshold $8 \, \sqrt{\frac{\pi^3}{3}}$, then the unique flow line of the Willmore flow, starting to move in $F_0$, converges fully to a conformal image of the standard Clifford-torus in every $C^{m}$-norm, up to time dependent, smooth reparametrizations. Key instruments for the proofs are the equivariance of the Hopf-fibration $\pi:\mathbb{S}^3 \to \mathbb{S}^2$ w.r.t. the effect of the $L^2$-gradient of the Willmore energy applied to smooth Hopf-tori in $\mathbb{S}^3$ and to smooth closed regular curves in $\mathbb{S}^2$, a particular version of the Lojasiewicz-Simon gradient inequality, and a certain mathematical bridge between the Euler-Lagrange equation of the elastic energy functional and a particular class of elliptic curves over $\mathbb{C}$

Unstable extremal surfaces of the "Shiffman functional” spanning rectifiable boundary curves

In this paper we derive a sufficient condition for the existence of extremal surfaces of a parametric functional $$\mathcal{J}$$ with a dominant area term, which do not furnish global minima of $$\mathcal{J}$$ within the class $$\mathcal{C}^*(\Gamma )$$ of H 1,2-surfaces spanning an arbitrary closed rectifiable Jordan curve $$\Gamma\subset \mathbb{R}^3$$ that merely has to satisfy a chord-arc condition. The proof is based on the "mountain pass result” of (Jakob in Calc Var 21:401-427, 2004) which yields an unstable $$\mathcal{J}$$ -extremal surface bounded by an arbitrary simple closed polygon and Heinz' ”approximation method” in (Arch Rat Mech Anal 38:257-267, 1970). Hence, we give a precise proof of a partial result of the mountain pass theorem claimed by Shiffman in (Ann Math 45:543-576, 1944) who only outlined a very sketchy and partially incorrect proo

Local boundedness of the number of solutions of Plateau's problem for polygonal boundary curves

In the present article, the author proves two generalizations of his "finiteness-result” (I.H.P. Anal. Non-lineaire, 2006, accepted) which states for any extreme simple closed polygon $$\Gamma \subset {\mathbb{R}}^3$$ that every immersed, stable disc-type minimal surface spanning Γ is an isolated point of the set of all disc-type minimal surfaces spanning Γ w.r.t. the C 0-topology. First, it is proved that this statement holds true for any simple closed polygon in $${\mathbb{R}}^3$$ , provided it bounds only minimal surfaces without boundary branch points. Also requiring that the interior angles at the vertices of such a polygon Γ have to be different from $$\frac{\pi }{2}$$ the author proves the existence of some neighborhood O of Γ in $${\mathbb{R}}^3$$ and of some integer $$\beta$$ , depending only on Γ, such that the number of immersed, stable disc-type minimal surfaces spanning any simple closed polygon contained in O, with the same number of vertices as Γ, is bounded by $$\beta$

Schwarz operators of minimal surfaces spanning polygonal boundary curves

This paper examines the Schwarz operator A and its relatives Ȧ, Ā and Ǡ that are assigned to a minimal surface X which maps consequtive arcs of the boundary of its parameter domain onto the straight lines which are determined by pairs P j , P j+1 of two adjacent vertices of some simple closed polygon $${\Gamma\subset \mathbb{R}^3}$$ . In this case X possesses singularities in those boundary points which are mapped onto the vertices of the polygon Γ. Nevertheless it is shown that A and its closure Ā have essentially the same properties as the Schwarz operator assigned to a minimal surface which spans a smooth boundary contour. This result is used by the author to prove in [Jakob, Finiteness of the set of solutions of Plateau's problem for polygonal boundary curves. I.H.P. Analyse Non-lineaire (in press)] the finiteness of the number of immersed stable minimal surfaces which span an extreme simple closed polygon Γ, and in [Jakob, Local boundedness of the set of solutions of Plateau's problem for polygonal boundary curves (in press)] even the local boundedness of this number under sufficiently small perturbations of

Functional analytic properties and regularity of the M\"obius-invariant Willmore flow in R3\mathbb{R}^3

In this article we continue the author's investigation of the M\"obius-invariant Willmore flow moving parametrizations of umbilic-free tori in $\mathbb{R}^3$ and in the $3$-sphere $\mathbb{S}^3$. In the main theorems of this article we prove basic properties of the evolution operator of the "DeTurck modification" of the M\"obius-invariant Willmore flow and of its Fr\'echet derivative by means of a combination of the author's results about the conformally invariant Willmore flow with the theory of "bounded $\mathcal{H}_{\infty}$-calculus" for linear elliptic operators due to Amann, Denk, Hieber, Pr\"uss and Simonett, and with Amann's and Lunardi's work on semigroups and interpolation theory. Precisely, we prove local real analyticity of the evolution operator $[F \mapsto \mathcal{P}^*(\,\cdot\,,0,F)]$ of the "DeTurck modification" of the M\"obius-invariant Willmore flow in a small open ball in $W^{4-\frac{4}{p},p}(\Sigma,\mathbb{R}^3)$, for any $p>5$, about any fixed real analytic parametrization $F_0:\Sigma \longrightarrow \mathbb{R}^3$ of a compact, analytic and umbilic-free torus in $\mathbb{R}^3$. We prove moreover that the flow line $\{\mathcal{P}^*(\,\cdot\,,0,F_0)\}$, starting to move in such a umbilic-free immersion $F_0$, is real analytic on $\Sigma \times (0,T_{max}(F_0))$, and that the Fr\'echet derivative $D_{F}\mathcal{P}^*(\,\cdot\,,0,F_0)$ of the evolution operator in $F_0$ can be uniquely extended to a family of continuous linear operators $G^{F_0}(t_2,t_1)$ in $L^p(\Sigma,\mathbb{R}^3)$, whose ranges are dense in $L^{p}(\Sigma,\mathbb{R}^3)$, for every fixed pair of times $t_2 \geq t_1$ within $(0,T_{max}(F_0))$

Global existence and full convergence of the M\"obius-invariant Willmore flow in the 33-sphere

In this article we prove a ``global existence and full smooth convergence theorem'' for flow lines of the M\"obius-invariant Willmore flow, and we use this result, in order to prove that fully and smoothly convergent flow lines of the M\"obius-invariant Willmore flow are stable w.r.t. perturbations of their initial immersions in any $C^{4,\gamma}$-norm, provided they converge to a smooth parametrization of ``a Clifford-torus'' in $\mathbb{S}^3$. More precisely, in our stability result we prove that for every $C^{\infty}$-smooth initial immersion $F_0:\Sigma \longrightarrow \mathbb{S}^3$, whose corresponding unique flow line of the M\"obius-invariant Willmore flow converges fully and smoothly to a diffeomorphic parametrization of ``a Clifford-torus'' in $\mathbb{S}^3$, and for every fixed $k\in \mathbb{N}$ and $\gamma \in (0,1)$, there is a small open ball $B_{r}(F_0)$ in $C^{4,\gamma}(\Sigma,\mathbb{R}^4)$ about the immersion $F_0$, such that for any $C^{\infty}$-smooth initial immersion $F:\Sigma \longrightarrow \mathbb{S}^3$ in $B_{r}(F_0)$ the corresponding flow line $\{\mathcal{P}(\,\cdot\,,0,F)\}$ of the M\"obius-invariant Willmore flow exists globally and converges - up to smooth reparametrization - fully in $C^k(\Sigma,\mathbb{R}^4)$ to a smooth immersion, which parametrizes ``a Clifford-torus'' in $\mathbb{S}^3$. The proof relies on a combination of the author's achievements and techniques of his recent papers - treating the M\"obius-invariant Willmore flow - with Escher's, Mayer's and Simonett's work from ``the 90ies'' 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 $\mathbb{R}^3$