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Variational Theory and Spectral Theory of the Volume Functional

体积泛函的变分理论和谱理论

基本信息

批准号：
1710846
负责人：
Andre Neves
金额：
$ 30万
依托单位：
University of Chicago
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1710846&HistoricalAwards=false
关键词：
Variational Theory Spectral Volume Functional

项目摘要

Award: DMS 1710846, Principal Investigator: Andre NevesThe study of shapes that are in an equilibrium position was started by Lagrange 300 year ago. These special shapes are called minimal surfaces and they are ubiquitous in Science, serving to model soap films, black holes in General Relativity, or tensile structures in architecture. Moreover, they come in two types: those which are stable, i.e., if perturbed they return to their original equilibrium position and those which are unstable, i.e., if perturbed they move away from their original position. The first type has been extensively studied in Mathematics over the last 40 years and they have been used to solve many long standing open questions in Geometry and Mathematical Relativity. The second type started being studied more systematically some years ago by Marques and the PI and they were used to solve some open problems such as the Willmore Conjecture or the problem of finding infinitely many unstable minimal surfaces. The project presented plans to continue the study of unstable minimal surfaces. Its goals are twofold: On one hand to develop the theory that governs their existence and on the other hand to study their properties as the degrees of instability become quite large because this is expected to uncover new relations across Geometry, Topology, and Analysis.More precisely, the objectives of the project are to further develop the Almgren-Pitts Min-max Theory and to study the properties of minimal surfaces when seen as nonlinear eigenvalues to the volume spectrum. For the first part we aim to investigate how to bound from below the index of min-max minimal surfaces by the number of parameters and to relate that problem with the multiplicity one conjecture. An application of that result would be to solve a stronger version of a conjecture of Yau. The second objective is motivated by the recent Weyl Law for the volume spectrum that the PI proved with Marques and Liokumovich. This property suggests a stronger analogy between eigenfunctions for the Laplacian spectrum and minimal surfaces that we intend to explore and the possibility to prove Weyl Laws for many other nonlinear problems. The techniques involved combine ideas from Spectral Geometry, Morse Theory, and Minimal Surface theory.

奖项：DMS 1710846，首席研究员：安德烈·内夫斯对处于平衡位置的形状的研究始于300年前的拉格朗日。这些特殊的形状被称为最小表面，它们在科学中无处不在，用于模拟肥皂膜，广义相对论中的黑洞或建筑中的拉伸结构。此外，它们有两种类型：稳定的，即，如果受到扰动，则它们返回到它们的原始平衡位置，而那些不稳定的，即，如果受到扰动，则它们会离开其原始位置。在过去的40年里，第一种类型在数学中得到了广泛的研究，它们被用来解决几何和数学相对论中许多长期存在的开放性问题。第二种类型开始被更系统地研究几年前由马克斯和PI和他们被用来解决一些开放的问题，如Willmore猜想或问题，找到无穷多个不稳定的极小曲面。该项目提出了继续研究不稳定极小曲面的计划。其目标有两个：一方面发展理论，支配他们的存在，另一方面研究他们的性质，因为不稳定的程度变得相当大，因为这有望揭示新的关系，几何，拓扑和分析。更确切地说，该项目的目标是进一步发展阿尔姆皮茨矿业，max理论，研究极小曲面作为体谱非线性特征值时的性质。对于第一部分，我们的目标是研究如何从下面的指标的最小最大极小曲面的参数的数量，并与该问题的多重性一猜想。这个结果的一个应用是解决丘的一个猜想的更强版本。第二个目标是由最近的外尔定律的体积谱，PI证明与马克斯和Liokumovich。这一性质表明，拉普拉斯谱和极小曲面的本征函数之间有更强的相似性，我们打算探索和证明许多其他非线性问题的外尔定律的可能性。所涉及的技术联合收割机的想法，从光谱几何，莫尔斯理论，最小曲面理论。