Variational Theory and Spectral Theory of the Volume Functional

奖项：DMS 1710846，首席研究员：Andre neves对处于平衡位置的形状的研究是300年前由拉格朗日开始的。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：一种是稳定的，即如果受到干扰，它们会回到原来的平衡位置；另一种是不稳定的，即如果受到干扰，它们会远离原来的位置。在过去的40年里，第一种类型在数学领域得到了广泛的研究，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。第二种类型在几年前开始被Marques和PI更系统地研究它们被用来解决一些开放的问题，比如Willmore猜想或者寻找无穷多个不稳定最小曲面的问题。该项目提出了继续研究不稳定最小表面的计划。它的目标有两个：一方面是发展管理它们存在的理论，另一方面是研究它们在不稳定程度变得相当大时的性质，因为这有望揭示几何、拓扑和分析之间的新关系。更准确地说，该项目的目标是进一步发展Almgren-Pitts最小-最大理论，并研究当被视为体积谱的非线性特征值时最小表面的性质。在第一部分中，我们的目的是研究如何用参数数从下限定最小-最大最小曲面的索引，并将该问题与多重性1猜想联系起来。该结果的一个应用将是解决丘猜想的一个更强的版本。第二个目标是由最近的Weyl定律驱动的体积谱，PI与Marques和Liokumovich证明了这一点。这一性质表明拉普拉斯谱的本征函数与我们将要探索的极小曲面之间有更强的相似性，并且有可能为许多其他非线性问题证明Weyl定律。所涉及的技术结合了光谱几何、莫尔斯理论和最小曲面理论的思想。

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.