Curvature and Metric Measure Geometry

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Abstract for DMS - 0104128This project is concerned with the structure of smoothly curved spaceson a small but definite scale, with the partial reularity and singularitystructure of limit cases of such spaces, with the relations between analysisand geometry on metric measure spaces, and with the structure of nonpositively curved spaces. Specifically, we will continue our investigationsof manifolds with Ricci curvature bounded below and their weak limits,of Einstein manifolds and their weak limits, of metric measure spacesfor which a doubling condition and Poincar\'e inequality hold, and ofcompact nonpositively curved manifolds whose volume is sufficiently small.Smoothly curved spaces (the higher dimensional generalizations of curvedsurfaces) play an important role in geometry and in physics.A central issue is to describe the ``worst possible'' examples of spacesthe 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''.