Variational Principles for immersed Surfaces with $L^2$-bounded Second Fundamental Form
Variational Principles for immersed Surfaces with $L^2$-bounded Second Fundamental Form
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具有 $L^2$ 有界第二基本形式的浸没曲面的变分原理
DOI：
10.1515/crelle-2012-0106
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
T. Rivière
中科院分区：
文献类型：
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作者：
T. Rivière
在这项工作中，我们提出了新的基本工具，用于研究浸没表面的 Willmore 泛函到 $R^m$ 的变化。例如，这种方法给出了任意余维中任意闭合曲面的威尔莫尔最小化嵌入存在性的新证明。我们解释了相同的方法如何解决 Willmore 泛函的约束最小化问题。我们特别表明，对于给定的封闭曲面和给定的该曲面的共形类，$R^m$ 中存在浸入，可能远离孤立的分支点，这使得 $R^m$ 中所有可能的 Lipschitz 浸入中的威尔莫尔能量最小化，具有 $L^2-$ 有界第二基本形式并实现了该共形类。该分支浸没是平滑保形Willmore分支浸没或等温分支浸没。我们证明，只要共形类中的最小能量小于 $8\pi$，分支点就不存在，并且在这种情况下，这些浸没扩展到平滑的共形 Willmore 嵌入或表面的全局等温嵌入。最后，作为我们分析的副产品，我们确定在模空间的紧致子空间内，以下成立：Palais Smale Willmore 序列的弱极限是共形 Willmore，共形 Willmore 的 Palais Smale 序列的弱极限是共形 Willmore 或全局等温，最后我们还观察到全局等温沉浸的弱收敛 Palais Smale 序列是全局等温的。论文中进行的分析 - 特别是最后的结果 - 打开了使用最小最大方法在没有或有约束的情况下构造 Willmore 函数的新临界鞍点的可能性之门
In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into $R^m$. This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension. We explain how the same approach can solve constraint minimization problems for the Willmore functional. We show in particular that, for a given closed surface and a given conformal class for this surface, there is an immersion in $R^m$, away possibly from isolated branched points, which minimizes the Willmore energy among all possible Lipschitz immersions in $R^m$ having an $L^2-$bounded second fundamental form and realizing this conformal class. This branched immersion is either a smooth Conformal Willmore branched immersion or an isothermic branched immersion. We show that branched points do not exist whenever the minimal energy in the conformal class is less than $8\pi$ and that these immersions extend to smooth conformal Willmore embeddings or global isothermic embeddings of the surface in that case. Finally, as a by-product of our analysis, we establish that inside a compact subspace of the moduli space the following holds : weak limit of Palais Smale Willmore sequences are Conformal Willmore, that weak limits of Palais Smale sequences of Conformal Willmore are either Conformal Willmore or Global Isothermic and finally we observe also that weakly converging Palais Smale sequences of Global Isothermic Immersions are Global Isothermic. The analysis developped along the paper - in particular these last results - opens the door to the possibility of constructing new critical saddle points of the Willmore functional without or with constraints using min max methods
固定共形类下 Willmore 泛函的极小化
DOI：
10.4310/jdg/1361844942
发表时间：
2013
期刊：
arXiv: Differential Geometry
影响因子：
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作者：
Kuwert;Schätzle
通讯作者：
Schätzle