GEOMETRIC ANALYSIS ON ALEXANDROV SPACES

摘要

An Alexandrov space is a metric space with an (upper or lower) curvature bound. This kind of space was first introduced and studied by A. Wald [53] in 1935.1 However, Wald's definition of the boundedness of curvature was hard to deal with and there was no attractive progress on it. In 1951, A. D. Alexandrov~2 gave a clear definition of the boundedness of curvature using triangles ([1]) and studied Alexandrov spaces in the two-dimensional case comprehensively ([2]).~3 Starting from Alexandrov's definition, Alexandrov spaces are studied by Alexandrov himself and his followers ceaselessly. In the 1980s, M. Gromov and others began to study convergence or collapsing of Riemannian manifolds, in which the importance of Alexandrov spaces was recognized. Recently, Alexandrov spaces have been used in G. Perelman's proof of the geometrization conjecture ([37, 46]).