The Asymptotically Flat Scalar-Flat Yamabe Problem with Boundary
We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with inner boundary in dimension \(n\ge 3\). First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric g, there is a conformally equivalent asymptotically flat scalar-flat metric that agrees with g on the boundary. We then replace the metric boundary condition with a condition on the mean curvature: given a function f on the boundary that is not too large, we show that there is an asymptotically flat scalar-flat metric, conformally equivalent to g whose boundary mean curvature is given by f. The latter case involves solving an elliptic PDE with critical exponent using the method of sub- and supersolutions. Both results require the usual assumption that the Sobolev quotient is positive.
KeywordsYamabe problem Asymptotically flat manifold Scalar curvature
Mathematics Subject Classification58J05 53C21 53A30 35B33
- 3.Bartnik, R., Isenberg, J.: The constraint equations. In: Chruściel, P., Friedrich, H. (eds.) The Einstein Equations and the Large Scale Behavior of Gravitational Fields, pp. 1–38. Birkhäuser, Basel (2004)Google Scholar
- 4.Brendle, S., Marques, F.C.: Recent progress on the Yamabe problem. arXiv:1010.4960 (2010)
- 14.McCormick, S.: The hilbert manifold of asymptotically flat metric extensions. arXiv:1512.02331 (2015)