Willmore surfaces in Riemannian manifolds

黎曼流形中的威尔莫尔曲面

• 批准号: 245965278
• 负责人: Professor Dr. Tobias Lamm
• 金额: --
• 依托单位: Institut für Analysis
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2019-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/245965278?language=en
• 关键词: Willmore; surfaces; Riemannian; manifolds

Geometric variational problems arise naturally in various branches of mathematics, physics, biology and computer science. The most prominent examples are the isoperimetric problem and minimal surfaces. In this project we study higher order variational problems such as the Willmore functional and variants thereof. This specific functional arises for example in general relativity, biology and in the theory of image restoration. In the last few years the Willmore functional and its variants were extensively studied in Euclidean space and our goal here is to develop and partially extend the theory to arbitrary target manifolds since this is relevant in the above mentioned applications. More precisely, we want to study the effect of the ambient curvature on the geometric and analytical properties of the functionals. Our main motivation comes from applications to general relativity and we want to better understand the relation between the Willmore functional and the relevant physical quantities in this theory. In order to achieve our goals we have to extend the existing regularity, compactness and existence results for the functionals under consideration and since we are dealing with critical problems this requires a careful and delicate study of the underlying partial differential equations.

几何变分问题在数学、物理学、生物学和计算机科学的各个分支中自然产生。最突出的例子是等周问题和极小曲面。在这个项目中，我们研究高阶变分问题，如Willmore功能及其变种。这个特定的泛函出现在例如广义相对论、生物学和图像恢复理论中。在过去的几年中，Willmore泛函及其变体在欧氏空间中得到了广泛的研究，我们的目标是将理论发展并部分扩展到任意目标流形，因为这与上述应用有关。更准确地说，我们想研究的效果上的几何和分析性质的泛函的环境曲率。我们的主要动机来自于广义相对论的应用，我们希望更好地理解Willmore泛函与该理论中相关物理量之间的关系。为了实现我们的目标，我们必须延长现有的正则性，紧凑性和存在性的考虑中的泛函的结果，因为我们正在处理的关键问题，这需要一个仔细和微妙的研究基础偏微分方程。

项目成果

Rigidity and non-rigidity results for conformal immersions

保形浸入的刚性和非刚性结果

• DOI: 10.1016/j.aim.2015.06.006
• 发表时间: 2015
• 期刊: arXiv: Differential Geometry
• 作者: T. Lamm;R. M. Schätzle
• 通讯作者: R. M. Schätzle

Optimal rigidity estimates for nearly umbilical surfaces in arbitrary codimension

任意余维近脐表面的最佳刚度估计

• DOI: 10.1007/s00039-014-0303-6
• 发表时间: 2014
• 期刊: Geometric and Functional Analysis
• 影响因子: 2.2
• 作者: T. Lamm;R. M. Schätzle
• 通讯作者: R. M. Schätzle

Isoperimetric structure of asymptotically conical manifolds

渐近圆锥流形的等周结构

• DOI: 10.4310/jdg/1483655857
• 发表时间: 2017
• 期刊: arXiv: Differential Geometry
• 作者: O. Chodosh;M. Eichmair;A. Volkmann
• 通讯作者: A. Volkmann

Conformal Willmore tori in ℝ4

â4 中的保形威尔莫尔托里

• DOI: 10.1515/crelle-2015-0101
• 发表时间: 2018
• 期刊: Journal für die reine und angewandte Mathematik (Crelles Journal)
• 作者: T. Lamm;R. M. Schätzle
• 通讯作者: R. M. Schätzle

A note on Willmore minimizing Klein bottles in Euclidean space

关于威尔莫尔在欧几里得空间中最小化克莱因瓶的注释

• DOI: 10.1016/j.aim.2017.08.021
• 发表时间: 2017
• 期刊: arXiv: Differential Geometry
• 作者: J. Hirsch;E. Mäder-Baumdicker
• 通讯作者: E. Mäder-Baumdicker