Curvature and Metric Measure Geometry

DMS -0104128的摘要本项目涉及小而确定尺度上光滑弯曲空间的结构，这种空间极限情况的部分正则性和奇异性结构，度量测度空间上分析与几何之间的关系，以及非正弯曲空间的结构。 具体来说，我们将继续我们的研究与Ricci曲率下有界的流形和它们的弱极限，爱因斯坦流形和它们的弱极限，度量测度空间的双重条件和庞加莱不等式成立，和体积足够小的紧致非正曲流形。光滑曲空间（曲面的高维推广）在几何学和物理学中起着重要的作用。最坏的可能”空间的例子曲率以某种特定的方式控制，并且， 这是非常密切相关的，来描述在极限情况下可能形成的奇点。这个研究是我们项目的主要关注点。一个例子（从一个稍微不同的领域），我们的意思是“奇点的形成”，人们可以想到"黑洞“的形成。

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''.