arXiv subjects

Ruben Jakob

Publications and source records attributed to Ruben Jakob.

7 recordsLinked to original sources

arXiv2026

Uniqueness of Finite-Time Varifold Limits for the M\"obius-Invariant Willmore Flow

We prove uniqueness of finite-time geometric endpoints for the M\"obius-invariant Willmore flow in $\mathbb{S}^3$ under uniform quantitative nonumbilicity. The multiplicity-counting varifolds converge, without reparametrization or M\"obius renormalization, to a unique integral two-varifold. An intrinsic transport estimate gives quantitative total-variation convergence of the induced area measures on the fixed domain and bounded-Lipschitz Cauchy control of their pushforwards. Together with Allard compactness and rectifiability, this upgrades subsequential compactness to full-trajectory varifold convergence. The limit has generalized Euclidean mean curvature in $L^2$ with the natural endpoint lower-semicontinuity bound. For finite maximal trajectories with initial energy at most $8\pi$, Jakob's subsequential alternative becomes sequence independent: the limit is zero, or it has unit density and embedded Lipschitz support of genus zero or one. For Hopf-torus trajectories under the same energy bound, nonumbilicity is automatic and the anchored constant-speed profiles converge weakly in $W^{2,2}$ and strongly in $W^{1,2}$ and $C^{1,\alpha}$ for every $\alpha<\frac{1}{2}$. At infinite time, the same method yields a unique limit under an additional finite-dissipation-length condition.

math.AP↗

arXiv2026

Corrections to: "The Willmore flow of Hopf-tori in the 3-sphere"

This erratum addresses a logical mistake in the author's article [Jakob, R. The Willmore flow of Hopf-tori in the $3$-sphere. Journal of Evolution Equations 23, No. 72 (2023)] which resulted in two wrong assertions in parts (II) and (III) of Theorem 1 in the author's cited paper and in an inaccuracy in the formulation of the second part of Proposition 6 in that paper. We will not only point out these mistakes and their corrections, but we will additionally give a concrete counterexample to Statement (III) in Theorem 1 of the author's cited article which will automatically imply the optimality of the corrected version of that statement, as it is formulated in this erratum.

math.AP↗

arXiv2022

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

Here we continue the investigation of the M\"obius-invariant Willmore flow (MIWF), starting to move in arbitrary smooth and umbilic-free initial immersions $F_0$ which map some fixed compact torus $\Sigma$ into $\mathbb{R}^n$ respectively $\mathbb{S}^n$. Here we investigate the behaviour of flow lines $\{F_t\}$ of the MIWF in $\mathbb{S}^3$ starting with relatively low Willmore energy, as the time $t$ approaches the maximal time of existence $T_{max}(F_0)$ of $\{F_t\}$. 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 $\{F_t\}$ of the MIWF can be identified with the support of an integral $2$-varifold $\mu$ in $\mathbb{R}^4$, which is the weak limit of the sequence of varifolds $\{\mathcal{H}^2\lfloor_{F_{t_{l}}(\Sigma)}\}$, for an appropriately chosen sequence $t_{l} \nearrow T_{max}(F_0)$, and that $spt(\mu)$ is either empty or homeomorphic to some compact, closed manifold of genus either $0$ or $1$. In the particular case in which $spt(\mu)$ is a compact surface of genus $1$ it can be parametrized by a uniformly conformal bi-Lipschitz homeomorphism $f$ of class $W^{2,2}\cap W^{1,\infty}$, and under certain additional conditions on $\{F_{t_{l}}\}$ such a parametrization is a diffeomorphism of class $W^{4,2}$. Finally, if the initial immersion $F_0$ of a flow line $\{F_t\}$ is assumed to parametrize a Hopf-torus in $\mathbb{S}^3$ with Willmore energy not bigger than $8 \pi$, then we obtain more precise statements about the flow line $\{F_t\}$ as $t \nearrow T_{max}(F_0)$. 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 M\"obius-transformations of $\mathbb{S}^3$ - as $t \nearrow \infty$.

math.DG↗

arXiv2021

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

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.

math.DG↗

arXiv2020

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

In this article we continue the author's investigation of the M\"obius-invariant Willmore flow moving parametrizations of umbilic-free tori in $\mathbb{R}^n$ and in the $n$-sphere $\mathbb{S}^n$. 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 this topic with the theory of "bounded $\mathcal{H}_{\infty}$-calculus" for linear elliptic operators due to Amann, Denk, Duong, 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 $W^{4-\frac{4}{p},p}(\Sigma,\mathbb{R}^n)$, for any $p\in (3,\infty)$, about any fixed smooth parametrization $F_0:\Sigma \longrightarrow \mathbb{R}^n$ of a compact and umbilic-free torus in $\mathbb{R}^n$. We prove moreover that the entire maximal flow line $\mathcal{P}^*(\,\cdot\,,0,F_0)$, starting to move in a smooth and umbilic-free initial immersion $F_0$, is real analytic for positive times, and that therefore 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 $L^p(\Sigma,\mathbb{R}^n)$, whose ranges are dense in $L^{p}(\Sigma,\mathbb{R}^n)$, for every fixed pair of times $t_2\geq t_1$ within the interval of maximal existence $(0,T_{max}(F_0))$.

math.AP↗

arXiv2020

The Willmore flow of Hopf-tori in the $3$-sphere

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 $\mathbb{S}^3$. 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 or equal to the threshold $\frac{8\pi^2}{\sqrt{2}}$, then the unique flow line of the Willmore flow, starting to move in $F_0$, converges fully to a conformally transformed 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 $\pi:\mathbb{S}^3 \longrightarrow \mathbb{S}^2$ 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 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.

math.AP↗

arXiv2012

Immersed solutions of Plateau's problem for piecewise smooth boundary curves with small total curvature

We provide a new proof of the classical result that any closed rectifiable Jordan curve Gamma in space being piecewise of class C^2 bounds at least one immersed minimal surface of disc-type, under the additional assumption that the total curvature of Gamma is smaller than 6*Pi. In contrast to the methods due to Osserman, Gulliver and Alt, our proof relies on a polygonal approximation technique, using the existence of immersed solutions of Plateau's problem for polygonal boundary curves, provided by the first author's accomplishment of Garnier's ideas.

math.DG↗