Large Scale Geometry of Scalar Curvature and Minimal Surfaces

几何学家试图描述一个物体是如何弯曲的，并研究以特定方式弯曲的物体。曲率的研究在科学和工程的所有领域都很重要。例如，广义相对论假设重力以一种数学上精确的方式绘制空间和时间曲线，而在材料科学中，晶体结构之间的交汇点是由(相当不同的曲率概念)建模的。这个项目涉及到一种特殊的弯曲测量方法，称为标量曲率。标量曲率是最简单的弯曲度量之一，但由于这种简单性，标量曲率只能包含有限的信息量。因此，我们必须通过高度间接的方法来研究标量曲率。探索标量曲率的一种方法是与等周问题有关：在给定的空间中，我们如何以最小的周长包围最大的体积？这是最古老的数学问题之一，但它与标量曲率的联系直到最近才开始被理解。PI的项目将继续研究标量曲率，因为它影响面积和体积的大尺度行为，特别强调这些主题与与广义相对论相关的问题之间的关系。除了这项研究外，该项目还将通过研讨会组织、会议和暑期学校以及说明性文章和笔记，支持国际学生协会继续努力促进学生的学习和培训。研究计划的一个主要组成部分是继续研究大规模变分问题和标量曲率之间的联系，动机是几何和物理考虑，如彭罗斯不等式和广义相对论中的静态唯一性问题。为此，PI计划继续研究与标量曲率和等周问题有关的全局唯一性问题。最近，使用几何分析的强大工具的组合(其中许多被限制在三维)，已经在三维中理解了几个这样的问题。研究的一部分将包括研究这些结果的更高维度的类似物，这将需要开发一系列广泛的新技术。在上述全球独特性工作中形成的想法也导致了PI将调查的其他(先验无关的)主题。例如，确定Minkowski不等式对于非凸曲面(可能带有附加的弯曲项)的有效性与渐近平坦流形中的大稳定常平均曲率曲面的唯一性问题有关。类似地，对于某些具有非负数量曲率的黎曼三维流形，一个与环面同调类中最小面积相关的不变量与非负数量曲率三维流形中面积最小柱面的刚性有关。在一个不同的(但相关的)方向上，这个项目还将包括调查最小曲面的几何和拓扑之间的关系，包括研究具有简单拓扑或小指数的曲面。该奖项反映了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。

Geometers seek to describe how an object bends and study objects that curve in a specific way. This study of curvature is important in all domains of science and engineering. For example, the theory of general relativity posits that gravity curves space and time in a mathematically precise manner, while in materials science, the meeting points between crystal structures are modeled by (a rather different notion of) curvature. This project is concerned with the study of a particular measure of bending called scalar curvature. Scalar curvature is one of the simplest measures of bending, but due to this simplicity scalar curvature can contain only a limited amount of information. Hence, we must study scalar curvature through highly indirect means. One way to explore scalar curvature is in relation to the isoperimetric problem: in a given space, how can we enclose the largest amount of volume with the smallest perimeter? This is one of the oldest mathematical questions, but its link to scalar curvature is only recently beginning to be understood. The PI's project will continue the study of scalar curvature as it affects the large-scale behavior of area and volume, with particular emphasis on the relationship between such topics and problems related to general relativity. In addition to this research, this project will also support the PI's continued efforts to promote student learning and training through seminar organization, conferences, and summer schools, as well as expository articles and notes. A major component of the research plan is the continued study of the link between large-scale variational problems and scalar curvature, motivated by geometric and physical considerations such as the Penrose inequality and static uniqueness questions from general relativity. To this end, the PI plans to continue his investigation of global uniqueness questions related to scalar curvature and the isoperimetric problem. Recently, several such problems have been understood in three dimensions, using a combination of powerful tools from geometric analysis (many of which are limited to three dimensions). One portion of the research will consist of investigating higher dimensional analogues of these results, which will necessitate the development of a wide array of new techniques. The ideas developed in these aforementioned global uniqueness works have also led to other (a priori unrelated) topics that the PI will investigate. For example, determining the validity of the Minkowski inequality for non-convex surfaces (possibly with an additional bending term) is related to the uniqueness question for large stable constant mean curvature surfaces in asymptotically flat manifolds. Similarly, an invariant related to the least area in the homology class of a torus for certain Riemannian three-manifolds with non-negative scalar curvature is related to the rigidity of area-minimizing cylinders in three-manifolds of non-negative scalar curvature. In a different (but related) direction, this project will also include investigation of the relationship between the geometry and topology of minimal surfaces, including the study of surfaces with simple topology or small index.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.