Geometry, Analysis, and Variational Methods

这个项目将研究与最小曲面的变分理论及其应用有关的问题。最小表面是微分几何中最自然的物体之一，肥皂泡就是一个很好的例子。它们在许多领域都有应用，如三维拓扑学、数学物理、复杂和共形几何以及材料科学。在广义相对论中，最小表面是黑洞视界的模型。极小曲面方程作为几种非线性现象的模型起着非常重要的作用。最小表面最近也被应用于生物和化学领域的材料设计中。该项目还包括博士生和初级研究人员的培训。PI还将通过讲座、会议和讲习班传播他的作品。这个项目将推进我们对最小曲面及其一般存在理论的基本理解。它关注的基本问题是这些物体何时存在，以及它们的属性如何与周围环境的特征相关联。目的是研究给定黎曼流形中最小变分空间的莫尔斯理论性质。其思想是结合最小极大值方法与Almgren-Pitts最小极大值理论，以及具有同伦非平凡变种族存在的拓扑方法。PI将研究与此主题相关的几个问题，包括在高维和非紧化情况下的最小超曲面的构造。该项目还包括继续培养博士生和博士后研究人员的计划。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 will investigate questions related to the variational theory of minimal surfaces and its applications. Minimal surfaces, of which soap bubbles are an illustrative example, are among the most natural objects in differential geometry. They have applications in many areas, such as three-dimensional topology, mathematical physics, complex and conformal geometry, and materials science. In general relativity, minimal surfaces appear as models for the apparent horizons of black holes. The minimal surface equation plays a very important role as a model for several kinds of nonlinear phenomena. Minimal surfaces have also been recently used in the design of materials with applications in biology and in chemistry. The project also includes training of PhD students and junior researchers. The PI will also disseminate his work through lectures, conferences, and workshops.This project will advance our basic understanding of minimal surfaces and their general existence theory. It concerns foundational questions about when these objects exist and how their properties relate to features of the ambient. The aim is to investigate the Morse-theoretic properties of the space of minimal varieties in a given Riemannian manifold. The idea is to use a combination of min-max methods, with the Almgren-Pitts min-max theory, and topological methods with the existence of homotopically nontrivial families of varieties. PI will study several questions related to this theme, including constructions of minimal hypersurfaces in higher dimensions and in the noncompact case. The project also includes plans for continued training of PhD students and post-doctoral researchers.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.