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Nonexistence of Poincaré-Einstein Fillings on Spin Manifolds

Banff International Research Station for Mathematical Innovation and Discovery

In this talk, we discuss whether a conformal class on the boundary $M$ of a given compact manifold $X$ can be the conformal infinity of a Poincaré-Einstein metric in $X$. We construct an invariant of conformal classes on the boundary $M$ of a compact spin manifold $X$ of dimension $4k$ with the help of the Dirac operator. We prove that a conformal class cannot be the conformal infinity of a Poincaré-Einstein metric if this invariant is not zero. Furthermore, we will prove this invariant can attain values of infinitely many integers if one invariant is not zero on the above given spin manifold. This talk is based on a joint work with Gursky and Stolz.

Mathematics; Global analysis, analysis on manifolds; Relativity and gravitational theory; Global analysis

video/mp4

Author affiliation: University of Notre Dame

2019-03-22

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/69048

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/