$W^{2,2}$-conformal immersions of a closed Riemann~surface into $\R^n$
Ernst Kuwert,Yuxiang Li +1 more
TLDR
In this paper, a branched conformal immersion of compact Riemann surfaces with fixed genus and Willmore energy was studied, and it was shown that the map is unbranched under the assumption that π_k converges to π-Sigma in moduli space.Abstract:
We study sequences $f_k:\Sigma_k \to \R^n$ of conformally immersed, compact Riemann surfaces with fixed genus and Willmore energy ${\cal W}(f) \leq \Lambda$. Assume that $\Sigma_k$ converges to $\Sigma$ in moduli space, i.e. $\phi_k^\ast(\Sigma_k) \to \Sigma$ as complex structures for diffeomorphisms $\phi_k$. Then we construct a branched conformal immersion $f:\Sigma \to \R^n$ and M\"obius transformations $\sigma_k$, such that for a subsequence $\sigma_k \circ f_k \circ \phi_k \to f$ weakly in $W^{2,2}_{loc}$ away from finitely many points. For $\Lambda < 8\pi$ the map $f$ is unbranched. If the $\Sigma_k$ diverge in moduli space, then we show $\liminf_{k \to \infty} {\cal W}(f_k) \geq \min(8\pi,\omega^n_p)$. Our work generalizes results in \cite{K-S3} to arbitrary codimension.read more
Citations
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Journal ArticleDOI
Variational Principles for immersed Surfaces with L 2 -bounded Second Fundamental Form.
TL;DR: In this article, it was shown that branched points do not exist whenever the minimal energy in the conformal class is less than 8� and that these immersions extend to smooth conformal Willmore embeddings or global isothermic embedding of the surface in that case.
Journal ArticleDOI
Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow
Pierre Raphaël,Remi Schweyer +1 more
TL;DR: In this article, the authors considered the energy critical harmonic heat flow from a smooth compact revolution surface of a smooth manifold of codimension (L-1) to an energy critical topology and showed the existence of excited slow blow up rates and associated instability of these threshold dynamics.
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Energy quantization for Willmore surfaces and applications
Yann Bernard,Tristan Rivière +1 more
TL;DR: In this article, a bubble-neck decomposition for sequences of closed surfaces of a given genus modulo has been shown, together with an energy quantization result for sequences with uniformly bounded energy and nonsmooth conformal type.
Journal ArticleDOI
Lipschitz conformal immersions from degenerating Riemann surfaces with L2-boundedsecond fundamental forms
TL;DR: In this article, the authors give an asymptotic lower bound for the Willmore energy of weak immersions with degenerating conformal class, which is used in several other works.
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Energy Quantization for Willmore Surfaces and Applications
Yann Bernard,Tristan Rivière +1 more
TL;DR: In this paper, a bubbleneck decomposition of a set of closed surfaces of a given genus modulo the Mobius group action is shown to be compact under some energy threshold.
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TL;DR: In this paper, the authors introduce the concept of Riemannian geometry and present a general framework for the analysis of differential geometry with respect to differentially differentiable geometry.
Journal Article
Compensated compactness and Hardy spaces
Journal ArticleDOI
A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces
Peter Li,Shing-Tung Yau +1 more
TL;DR: The conformal volume of a compact Riemannian manifold with a fixed conformal structure was defined in this article, where it was shown that the conformal area of a manifold can be computed by the set of all branched conformal immersions obtained by composi t ion of qo with conformal automorphisms of S. In fact, if there exists a minimal immersion of M into S, where coordinate functions are first eigenfunctions, then the conformality of M is given by the area of M with respect to the induced metric.
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Boundary regularity and the Dirichlet problem for harmonic maps
Richard Schoen,Karen Uhlenbeck +1 more
TL;DR: In this article, it was shown that the boundary regularity of energy minimization maps with a prescribed Dirichlet boundary condition can be obtained in a full neighborhood of the boundary, assuming appropriate regularity on the manifold, the boundary and the data.
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