Department of Mathematics
Title : Smooth metric measure spaces and conformal geometry
| Speaker: Jeffrey Case Affiliation: Princeton University Time: 10am Monday, 19 March, 2012 Location: 412 |
Abstract
| Smooth metric measure spaces are (pseudo-)Riemannian manifolds $(M^n,g)$ together with a function $\phi$ and a parameter $m$ (usually nonnegative, but possible infinite) meant to specify that the measure $e^{-\phi}dvol_g$ is "$(m+n)$-dimensional." For example, such objects arise naturally when a family of $(m+n)$-dimensional manifolds collapses onto $M$. On such spaces, there are natural analogues of the Ricci and scalar curvature, and in particular it makes sense to talk about "quasi-Einstein" metrics, which include as special cases (conformally) Einstein metrics, static metrics (as time-symmetric hypersurfaces in a static spacetime), and gradient Ricci solitons. After making precise these definitions, we discuss how ideas from conformal geometry can be used to better understand smooth metric measure spaces and quasi-Einstein metrics. In particular, we talk about how one can generalize obstructions of Gover-Nurowski for conformally Einstein metrics and of Bartnik-Tod for static metrics to quasi-Einstein metrics. As time allows, we will also talk about the analogue of the conformal Laplacian on smooth metric measure spaces, and how it is useful in studying analytic questions involving quasi-Einstein metrics. |
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Programmes and Centres
- New Zealand Institute of Mathematics and its Applications (NZIMA)
- Community for Understanding and Learning in the Mathematical Sciences (CULMS)
- Centre for Mathematical Social Science (CMSS)
- Department of Computer Science
- Department of Engineering Science
- Department of Physics
- Department of Statistics
- Auckland Bioengineering Institute
