Sharp Eigenvalue Inequalities and Minimal Surfaces

本课题广泛涉及光谱几何，研究物体的某些解析不变量（称为光谱）与这些物体的形状（或几何形状）之间的关系。谱的一个经典例子是拉普拉斯谱。它是一个数字序列，编码了物体的许多物理特性的信息，如弹性、热量和声音传播等。更具体地说，这个项目处理特殊形状的光谱几何，如肥皂膜。肥皂膜的几何结构是现代几何分析的中心课题，其应用远远超出其名称所暗示的范围，最突出的是广义相对论的数学理论。肥皂膜的光谱几何是光谱理论与几何相结合的一个令人兴奋的新研究领域。这个领域固有的跨学科性质导致了光谱理论和几何分析之间富有成效的思想交流。该项目的目标是进一步发展近年来引入的技术和思想，研究它们在光谱几何之外的应用，并引起年轻研究人员对这一有前途的新学科的兴趣。该项目还包括本科生的暑期研究机会。本课题致力于流形上自然几何算子特征值的尖锐等周不等式的研究。该域的基本原理是这种不等式的最优度量与最小子流形之间的对应关系。因此，各种各样的方法可以应用于这个问题，并且该领域的进展通常会产生关于最小子流形几何的有趣信息。以下四个问题被确定为项目的重点。第一个目标是将PI在投影平面上所有拉普拉斯特征值的尖锐等周不等式的显式形式上的先前结果扩展到更复杂表面的情况下，从环面开始。第二个项目将研究最优度量的规律性和稳定性。这将通过与D. Stern合作获得的最优特征值不等式的能量最小-最大表征来实现。第三个目标是研究当边界分量的数量变得无界时，第一个Steklov特征值的最优度量的渐近行为。数值模拟结果表明，在这种情况下会出现一种新的几何现象：相应的自由边界最小曲面收敛于边界内的封闭最小曲面。目的是在理论上验证这一点，并进一步了解这种收敛的机制。最后，本课题将在拉普拉斯和斯特克洛夫特征值的框架下，发展狄拉克算子特征值的等周不等式理论。更广泛的影响包括共同组织会议和指导本科生进行暑期研究项目。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is broadly concerned with the subject of spectral geometry, which studies the relation between certain analytic invariants (called spectra) of objects and the shape (or geometry) of these objects. A classical example of a spectrum is a Laplacian spectrum. It is a sequence of numbers encoding the information about many physical properties of an object such as elasticity, heat and sound propagations and many others. More specifically, this project deals with the spectral geometry of special shapes such as soap films. The geometry of soap films is a central subject of modern geometric analysis with applications far beyond the one suggested by their name, most prominently to mathematical theory of general relativity. Spectral geometry of soap films is an exciting novel area of research on the interface of spectral theory and geometry. The inherent interdisciplinary nature of this field results in a fruitful exchange of ideas between spectral theory and geometric analysis. The goal of the project is to further develop the techniques and ideas introduced the recent years, to investigate their applications beyond spectral geometry and to interest young researchers in this new promising subject. The project also includes summer research opportunities for undergraduate students. The project is devoted to the study of sharp isoperimetric inequalities for eigenvalues of natural geometric operators on manifolds. The underlying principle of the field is the correspondence between optimal metrics for such inequalities and minimal submanifolds. As a result, a variety of methods can be applied to this problem and advances in the field often result in interesting information on the geometry of minimal submanifolds. The following four problems are identified as the focus of the project. The first goal is to extend PI’s prior results on the explicit form of the sharp isoperimetric inequality for all Laplacian eigenvalues on the projective plane to the case of more complicated surfaces, starting with a torus. The second project will study the regularity and stability of optimal metrics. This will be achieved using the energy min-max characterization of the optimal eigenvalue inequalities obtained in collaboration with D. Stern. The third goal is to investigate the asymptotic behavior of optimal metrics for the first Steklov eigenvalue as the number of boundary components becomes unbounded. Numerical simulations predict that a novel geometric phenomenon manifests itself in this regime: the corresponding free boundary minimal surfaces converge to a closed minimal surface in the boundary. The aim is to verify this theoretically and further understand the mechanisms of such convergence. Finally, the project will develop the theory of isoperimetric inequalities for eigenvalues of Dirac operator following the framework of Laplace and Steklov eigenvalues. Broader impacts include co-organizing conferences and mentoring undergraduate students in summer research projects.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.