Geometry, Analysis, and Variational Methods

本项目将研究与极小曲面变分理论及其应用有关的问题。极小曲面是微分几何中最自然的物体之一，肥皂泡就是其中的一个例证。它们在许多领域都有应用，如三维拓扑学、数学物理、复杂和保角几何以及材料科学。在广义相对论中，极小表面表现为黑洞视界的模型。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。最近，极小表面也被用于材料的设计，在生物学和化学中有应用。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、会议和研讨会来传播他的工作。这个项目将促进我们对极小曲面及其一般存在理论的基本理解。它涉及有关这些对象何时存在以及它们的属性如何与环境特征相关的基本问题。目的是研究给定黎曼流形中极小变差空间的Morse理论性质。其思想是结合使用极小极大方法与Almgren-Pitts极小极大理论，以及存在同伦非平凡变异族的拓扑方法。PI将研究与这一主题相关的几个问题，包括高维极小超曲面的构造和非紧情况下的极小超曲面的构造。该项目还包括继续培训博士生和博士后研究人员的计划。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.