Sobolov-Class Asymptotically Hyperbolic Manifolds and the Yamabe Problem

David Maxwell

We consider asymptotically hyperbolic manifolds whose metrics have Sobolev-class regularity. Building on prior work by Allen, Isenberg, Lee, and Allen-Stavrov in the Hölder category, we introduce two new function spacesfor metrics potentially having a large amount of interior differentiability measured in Sobolev scales, but whose regularity implies only a Hölder continuous conformal structure. We establish Fredholm theorems for elliptic operators arising from metrics in these families.To demonstrate utility of our methods, we solve the Yamabe problem in this category. As a special limiting case, we show that the asymptotically hyperbolic Yamabe problem is solvable so long as the metric admits a $W^{1,p}$ conformal compactification, with $p$ greater than the dimension of the manifold.(Joint with Paul T. Allen, Lewis & Clark and John M. Lee, University of Washington)

http://scgp.stonybrook.edu/video_portal/video.php?id=5327

Simons- Workshop: Recent Advances on Scalar Curvature Problems