DIFFERENTIAL GEOMETRY/PDE SEMINAR

Thursday, May 13, 2004 Padelford C-36 1:30 pm

Boundary regularity of conformally compact Einstein metrics

Jack Lee (UW)

A Riemannian metric on the interior of a compact manifold with boundary is called conformally compact if it can be conformally rescaled to a metric that extends continuously to the entire compact manifold with boundary. The restriction of the rescaled metric to the boundary yields a well-defined conformal structure on the boundary, called the “conformal infinity” of the given metric. Conformally compact Einstein metrics have recently played an increasingly important role in mathematical physics because they provide a concrete realization of the holography principle of string theory via the socalled “AdS/CFT correspondence.” In order to realize this correspondence, it is necessary to know that conformally compact Einstein metrics have suitable asymptotic expansions at the boundary. It has long been conjectured that every conformally compact Einstein metric with smooth conformal infinity has a smooth conformal compactification in even dimensions, and an infiniteorder asymptotic expansion in powers of the distance to the boundary and its logarithm in odd dimensions. Smoothness has recently been proved in dimension 4 by Michael Anderson (with corrections by Dylan Helliwell).

In this talk, I will describe a proof of this conjecture in all dimensions. This is joint work with Piotr T. Chruściel, Erwann Delay, and Dale Skinner.

For more information about this seminar, visit the DG/PDE Seminar Web page (from the Math Department home page, www.math.washington.edu, follow the link Seminars, Colloquia, and Conferences).

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