The Journal of Geometric Analysis

, Volume 27, Issue 3, pp 2269–2277 | Cite as

The Asymptotically Flat Scalar-Flat Yamabe Problem with Boundary

Article

Abstract

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.

Keywords

Yamabe problem Asymptotically flat manifold Scalar curvature 

Mathematics Subject Classification

58J05 53C21 53A30 35B33 

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Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.School of Science and TechnologyUniversity of New EnglandArmidaleAustralia
  2. 2.Institutionen för MatematikKungliga Tekniska HögskolanStockholmSweden

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