Comparison and Inverse Comparison Geometry

University Of California-Riverside, Riverside CA

Investigators

Abstract

In 1827, Gauss published his now famous Theorema Egregium, which roughly translates into English as Remarkable Theorem. The Theorema Egregium established a relationship between the way a surface curves inside of space and how its intrinsic geometry differs from that of the classical geometry of the plane, studied by Euclid. An immediate consequence of the Theorema Egregium is that it is impossible to make a map of the world without distorting some crucial aspects of its geometry. More startling was the generalization of Gauss’s ideas to higher-dimensional spaces by his student Riemann in the mid 19th century, which gave birth to the subject that is now known as Riemannian Geometry. While Riemannian Geometry and its generalizations have found wide-ranging applications, for example, in relativity, particle physics, and data science, answers to many basic, fundamental questions in the subject are enduring mysteries. This project is a two-pronged attack on some of these questions via what are known as Comparison Geometry and Inverse Comparison Geometry. Comparison Geometry is the branch of Global Riemannian Geometry that draws geometric and topological conclusions about a space with some constraint on its curvature by comparing it to a model space with constant curvature. Inverse Comparison Geometry concerns the opposite question; that is, which spaces admit Riemannian geometries that satisfy a given geometric constraint. The project will employ methods from both subjects to continue attacking problems that lie at the heart of Global Riemannian Geometry. The project also includes advising and mentoring of students, continued commitment to DEI initiatives and mathematical dissemination. This research centers on three basic problems: the Diffeomorphism Stability Question, the Pinching Problem in positive curvature, and the constructions of manifolds with almost nonnegative curvature. The Diffeomorphism Stability Question asks whether a Gromov-Hausdorff convergent sequence of Riemannian manifolds has a stable diffeomorphism type provided the sequence is noncollapsing and has a uniform lower sectional curvature bound. Perelman's stability theorem guarantees that such a sequence has a stable topological type, and a result of Kuwae, Machigashira, and Shioya guarantees a stable diffeomorphism type provided the limit is nonsingular, in the appropriate sense. A generic limit has singularities. Nevertheless, the PI and his collaborators Grove, Sill, and Pro, have established that certain singular limit spaces are di¤eomorphically stable. This includes a recent result by Pro and the PI showing that Diffeomorphism Stability holds in dimension 4, regardless of the singular structure of the limit space. Together with prior work by Grove-Petersen-Wu, Perelman, and Kirby-Siebenmann, this means that the conclusion of Cheeger's Finiteness Theorem holds without the hypothesis of the upper curvature bound. The project aims to establish (jointly with Chambers and Pro) that Piecewise Linear Stability holds in all dimensions. The project with Searle and Solórzano will develop tools to show that some limit spaces are not diffeomorphically stable. Searle, Solórzano, and the PI will also use these tools to show that there are almost nonnegatively curved exotic spheres in all dimensions 7 and higher that are congruent to 3 mod 4. The Pinching Problem in positive curvature asks how the topology of a positively curved manifold is constrained as its curvature is pinched. The gap between theorems and examples in this area is startling. The project in collaboration with Guijarro and Murphy will improve the Abresch-Meyer injectivity radius estimate by exploiting the Jacobi Field Comparison Lemma of Guijarro and the PI. The project also includes significant training and mentoring of students, mathematical dissemination and commitment to service and DEI initiatives. 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.