Curvature and Metric Measure Geometry

New York University, New York NY

Investigators

Abstract

Abstract for DMS - 0104128 This project is concerned with the structure of smoothly curved spaces on a small but definite scale, with the partial reularity and singularity structure of limit cases of such spaces, with the relations between analysis and geometry on metric measure spaces, and with the structure of nonpositively curved spaces. Specifically, we will continue our investigations of manifolds with Ricci curvature bounded below and their weak limits, of Einstein manifolds and their weak limits, of metric measure spaces for which a doubling condition and Poincar\'e inequality hold, and of compact nonpositively curved manifolds whose volume is sufficiently small. Smoothly curved spaces (the higher dimensional generalizations of curved surfaces) play an important role in geometry and in physics. A central issue is to describe the ``worst possible'' examples of spaces the curvature is controlled in some specific fashion, and, what is very closely related, to describe the kinds singularities which can form in limiting cases. This study is the main concern of our project. An example (from a somewhat different domain) of what we mean by the ``formation of singularities'', one can think of the formation of ``black holes''.