Talks

An extension of Perelman\'s soul theorem for singular spaces

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Jianguo Cao

2007-06-18
15:20:00 - 16:20:00

浦大邦講堂 , Institute of Atomic and Molecular Sciences, Dr. Poe Lecture Hall



In 1972, Cheeger and Gromoll conjectured that ``if an open complete smooth manifold Riemannian manifold M has non-negative curvature everywhere and positive curvature on an open ball, then M is diffeomorphic to an Euclidean space」. Using a new flat strip theorem, Perelman proved the Cheeger-Gromoll soul conjecture for smooth Riemannian manifolds. Perelman further proposed to study a similar problem for singular spaces. Using a new and completely different approach, we were able to able to derive a new soul theorem for singular spaces: ``Let M be an open, complete and finite dimensional Alexandrov space without boundary. Suppose that M has non-negative curvature everywhere and positive curvature on an open ball. Then M must be contractible」. Among other things, the speaker used various versions of the Riccati equation in the study of soul theorems. This is a joint work with B. Dai and J. Mei.