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Tobias Lamm

Publications and source records attributed to Tobias Lamm.

34 records · Page 2Linked to original sources
arXiv2015

Quantitative stratification and higher regularity for biharmonic maps

In this paper we prove quantitative regularity results for stationary and minimizing extrinsic biharmonic maps. As an application, we determine sharp, dimension independent $L^p$ bounds for $\nabla^k f$ that do not require a small energy hypothesis. In particular, every minimizing biharmonic map is in $W^{4,p}$ for all $1\le p<5/4$. Further, for minimizing biharmonic maps from $Ω\subset \mathbb{R}^5$, we determine a uniform bound on the number of singular points in a compact set. Finally, using dimension reduction arguments, we extend these results to minimizing and stationary biharmonic maps into special targets.

math.DG↗
arXiv2015

Compactness Results for Sequences of Approximate Biharmonic Maps

In this article, we prove energy quantization for approximate (intrinsic and extrinsic) biharmonic maps into spheres where the approximate map is in $L \log L$. Moreover, we demonstrate that if the $L\log L$ norm of the approximate maps does not concentrate, the image of the bubbles are connected without necks.

math.AP↗
arXiv2014

Optimal rigidity estimates for nearly umbilical surfaces in arbitrary codimension

In [dLMu05], DeLellis and Müller proved a quantitative version of Codazzi's theorem, namely for a smooth embedded surface $\ Σ\subseteq \mathbb{R}^3\ $ with area normalized to $\ {\cal H}^2(Σ) = 4 π $, it was shown that $\ \parallel A_Σ- id \parallel_{L^2(Σ)} \leq C \parallel A^0_Σ\parallel_{L^2(Σ)}\ $, and building on this, closeness of $\ Σ $ to a round sphere in $\ W^{2,2}\ $ was established, when $\ \parallel A^0_Σ\parallel_{L^2(Σ)}\ $ is small. This was supplemented in [dLMu06] by giving a conformal parametrization $\ S^2 \stackrel{\approx}{\longrightarrow} Σ $ with small conformal factor in $\ L^\infty\ $, again when $\ \parallel A^0_Σ\parallel_{L^2(Σ)}\ $ is small. In this article, we extend these results to arbitrary codimension. In contrast to [dLMu05], our argument is not based on the equation of Mainardi-Codazzi, but instead uses the monotonicity formula for varifolds.

math.DG↗
arXiv2014

Quantitative rigidity results for conformal immersions

In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in $\mathbb{R}^n$ with bounded total curvature. We show that (branched) conformal immersions which are close in energy to either a round sphere, a conformal Clifford torus, an inverted catenoid, an inverted Enneper's minimal surface or an inverted Chen's minimal graph must be close to these surfaces in the $W^{2,2}$-norm. Moreover, we apply these results to prove a corresponding rigidity result for complete, connected and non-compact surfaces.

math.DG↗
arXiv2012

Minimizers of the Willmore functional with a small area constraint

We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.

math.DG↗
arXiv2011

Branched Willmore Spheres

In this paper we classify branched Willmore spheres with at most three branch points (including multiplicity), showing that they may be obtained from complete minimal surfaces in $\R ^ 3$ with ends of multiplicity at most three. This extends the classification result of Bryant. We then show that this may be applied to the analysis of singularities of the Willmore flow of non-Willmore spheres with Willmore energy $ \mc {W} (f) \leq 16π$.

math.DG↗
arXiv2011

Two-dimensional curvature functionals with superquadratic growth

For two-dimensional, immersed closed surfaces $f:Σ\to \R^n$, we study the curvature functionals $\mathcal{E}^p(f)$ and $\mathcal{W}^p(f)$ with integrands $(1+|A|^2)^{p/2}$ and $(1+|H|^2)^{p/2}$, respectively. Here $A$ is the second fundamental form, $H$ is the mean curvature and we assume $p > 2$. Our main result asserts that $W^{2,p}$ critical points are smooth in both cases. We also prove a compactness theorem for $\mathcal{W}^p$-bounded sequences. In the case of $\mathcal{E}^p$ this is just Langer's theorem \cite{langer85}, while for $\mathcal{W}^p$ we have to impose a bound for the Willmore energy strictly below $8π$ as an additional condition. Finally, we establish versions of the Palais-Smale condition for both functionals.

math.AP↗
arXiv2011

Boundary regularity of stationary biharmonic maps

We consider the Dirichlet problem for stationary biharmonic maps $u$ from a bounded, smooth domain $Ω\subset\mathbb R^n$ ($n\ge 5$) to a compact, smooth Riemannian manifold $N\subset\mathbb R^l$ without boundary. For any smooth boundary data, we show that if, in addition, $u$ satisfies a certain boundary monotonicity inequality, then there exists a closed subset $Σ\subset\barΩ$, with $H^{n-4}(Σ)=0$, such that $u\in C^\infty(\barΩ\setminusΣ, N)$.

math.AP↗
arXiv2011

Geometric flows with rough initial data

We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence of a global unique and analytic solution to the Ricci-DeTurck flow on euclidean space for bounded initial metrics which are close to the euclidean metric in $L^\infty$ and to the harmonic map flow for initial maps whose image is contained in a small geodesic ball.

math.DG↗
arXiv2009

The heat flow with a critical exponential nonlinearity

We analyze the possible concentration behavior of heat flows related to the Moser-Trudinger energy and derive quantization results completely analogous to the quantization results for solutions of the corresponding elliptic equation. As an application of our results we obtain the existence of critical points of the Moser-Trudinger energy in a supercritical regime.

math.AP↗
arXiv2009

Foliations of asymptotically flat manifolds by surfaces of Willmore type

The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with nonzero mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch-Hawking mass. Thus our result has applications in the theory of General Relativity.

math.DG↗
arXiv2008

Energy identity for approximations of harmonic maps from surfaces

We prove the energy identity for min-max sequences of the Sacks-Uhlenbeck and the biharmonic approximation of harmonic maps from surfaces into general target manifolds. The proof relies on Hopf-differential type estimates for the two approximations and on estimates for the concentration radius of bubbles.

math.AP↗
arXiv2006

Conservation laws for fourth order systems in four dimensions

Following an approach of the second author for conformally invariant variational problems in two dimensions, we show in four dimensions the existence of a conservation law for fourth order systems, which includes both intrinsic and extrinsic biharmonic maps. With the help of this conservation law we prove the continuity of weak solutions of this system. Moreover we use the conservation law to derive the existence of a unique global weak solution of the extrinsic biharmonic map flow in the energy space.

math.AP↗