Geometry, analysis and variational methods

PI研究极小子流形的变分理论及其应用问题。极小曲面是微分几何中最自然的对象之一。它们在许多其他领域都有惊人的应用，如三维拓扑学、数学物理、复杂几何和保角几何等。在广义相对论中，极小表面作为黑洞视界的模型出现。极小曲面方程作为自然界中多种非线性现象的模型，起着非常重要的作用。这一领域的重大进展一直对数学分析和物理科学产生重大影响。本课题的研究将促进我们对极小曲面及其一般存在理论的基本理解。它涉及这些物体何时存在以及它们的特性如何与环境空间的特征相关的基本问题。这项研究项目包含了许多几何、分析和变分之间的交界点问题。该项目的目标之一是对给定黎曼流形中极小变差空间的Morse理论性质有一个很好的理解。这将通过极小极大技术和拓扑学方法的结合来实现，其中相关的圈空间是通过几何测度论来定义的。PI将研究最小-最大极小变差的存在性和基本性质，如Morse指数。

The PI investigates questions concerning the variational theory of minimal submanifolds and its applications. Minimal surfaces are among the most natural objects in differential geometry. They have encountered striking applications in many other fields, like three-dimensional topology, mathematical physics, complex and conformal geometry, among others. 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 in nature. Significant progress in this area has always had a great impact in mathematical analysis and the physical sciences. The research of 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 space. This research project contains a number of problems at the interface between geometry, analysis and the calculus of variations. One of the project's goals is to develop a good understanding of the Morse-theoretic properties of the space of minimal varieties in a given Riemannian manifold. This is to be accomplished by a combination of min-max techniques and topological methods, where the relevant spaces of cycles are defined by means of geometric measure theory. The PI will study the existence and basic properties, like the Morse index, of min-max minimal varieties.