Submanifolds and Foliations in Riemannian Manifolds

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Differential geometry is the study of the geometry of spaces. Sometimes these spaces exist as physical objects such as the surface of planet or a cell, or the universe itself; sometimes they appear as abstract spaces in applied situations, such as the set of all possible states of a physical system, the set of possible configurations of a machine, or the set of strategies of a group of players in the stock market. In these situations, it is often important to find and study the properties of objects inside these spaces that are optimal in some sense: for example, the shortest way to move between two configurations, or the surface that minimizes area or surface tension. Sometimes these geometric sub-objects come in families (e.g., foliations that slice the space into thin pieces), and it is important to study not just each of these pieces by itself, but the whole family as a unique entity. This project will study several instances of these important and very natural objects, and draw new bridges between these geometric concepts and other more abstract areas of mathematics. The main goal of the project is to study certain structures of Riemannian manifolds such as singular Riemannian foliations and closed minimal submanifolds, and relate them to the geometry and topology of the ambient space. In the study of singular Riemannian foliations, the main focus will be on laying new theoretical ground (e.g., continuing the development of an invariant theory in this context) and applying these concepts in different areas, such as producing new manifolds with positive and non-negative curvature, or exhibiting new immersed minimal hypersurfaces in round spheres and projective spaces. In the study of closed minimal submanifolds, the main focus will be on the index of such objects, especially in the two extreme cases of closed geodesics and minimal hypersurfaces. The PI will continue working on a conjecture of Berger on manifolds with all geodesics closed, and on a conjecture of Marques-Neves-Schoen on minimal hypersurfaces in manifolds with positive Ricci curvature. In the first case, the plan is to improve the understanding of the equivariant Morse theory on the free loop space of spaces with all geodesics closed. In the second case, the PI will continue to develop tools such as virtual immersions, recently developed by the PI, which could have independent interest. 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.