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The Asymptotically Flat Scalar-Flat Yamabe Problem with Boundary
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2017 (English)In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 27, no 3, p. 2269-2277Article in journal (Refereed) Published
Abstract [en]

We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with inner boundary in dimension n >= 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.

Place, publisher, year, edition, pages
SPRINGER , 2017. Vol. 27, no 3, p. 2269-2277
Keywords [en]
Yamabe problem, Asymptotically flat manifold, Scalar curvature
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-211389DOI: 10.1007/s12220-017-9760-0ISI: 000404679100022Scopus ID: 2-s2.0-85011281153OAI: oai:DiVA.org:kth-211389DiVA, id: diva2:1130155
Note

QC 20170808

Available from: 2017-08-08 Created: 2017-08-08 Last updated: 2017-08-08Bibliographically approved

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  • Other locale
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