Large Scale Geometry of Scalar Curvature and Minimal Surfaces

几何学试图描述物体如何弯曲，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数学上精确的方式弯曲空间和时间，而在材料科学中，晶体结构之间的交汇点是由曲率（一个相当不同的概念）建模的。这个项目是关于弯曲的一个特殊的措施称为标量曲率的研究。标量曲率是弯曲的最简单的度量之一，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与等周问题有关：在给定的空间中，我们如何用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广义相对论相关问题之间的关系。除了这项研究之外，该项目还将支持PI继续努力，通过研讨会组织，会议和暑期学校以及临时文章和笔记来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，其动机是几何和物理方面的考虑，如彭罗斯不等式和广义相对论的静态唯一性问题。为此，PI计划继续研究与标量曲率和等周问题相关的全局唯一性问题。最近，几个这样的问题已经被理解在三维，使用强大的工具，从几何分析（其中许多是有限的三维）的组合。研究的一部分将包括调查这些结果的高维类似物，这将需要开发一系列新技术。在上述全球唯一性工作中开发的想法也导致了PI将调查的其他（先验无关）主题。例如，确定Minkowski不等式对非凸曲面（可能有额外的弯曲项）的有效性与渐近平坦流形中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流形，与环面的同调类中的最小面积相关的不变量与具有非负数量曲率的三流形中的面积最小圆柱的刚性相关。在一个不同的（但相关的）方向，该项目还将包括最小曲面的几何和拓扑之间的关系的调查，包括简单拓扑或小指数的表面的研究。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.