Large Scale Geometry of Scalar Curvature and Minimal Surfaces

Stanford University, Stanford CA

Investigators

Abstract

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.