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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Prescribing curvatures on three dimensional Riemannian manifolds with boundaries


Author: Lei Zhang
Journal: Trans. Amer. Math. Soc. 361 (2009), 3463-3481
MSC (2000): Primary 35J60, 53B20
Published electronically: February 23, 2009
MathSciNet review: 2491888
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Abstract: Let $ (M,g)$ be a complete three dimensional Riemannian manifold with boundary $ \partial M$. Given smooth functions $ K(x)>0$ and $ c(x)$ defined on $ M$ and $ \partial M$, respectively, it is natural to ask whether there exist metrics conformal to $ g$ so that under these new metrics, $ K$ is the scalar curvature and $ c$ is the boundary mean curvature. All such metrics can be described by a prescribing curvature equation with a boundary condition. With suitable assumptions on $ K$,$ c$ and $ (M,g)$ we show that all the solutions of the equation can only blow up at finite points over each compact subset of $ \bar M$; some of them may appear on $ \partial M$. We describe the asymptotic behavior of the blow-up solutions around each blow-up point and derive an energy estimate as a consequence.


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Additional Information

Lei Zhang
Affiliation: Department of Mathematics, University of Alabama at Birmingham, 1300 University Boulevard, 452 Campbell Hall, Birmingham, Alabama 35294-1170
Email: leizhang@math.uab.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04911-3
PII: S 0002-9947(09)04911-3
Keywords: Scalar curvature, mean curvature, Harnack inequality
Received by editor(s): September 13, 2006
Published electronically: February 23, 2009
Additional Notes: The author was supported by National Science Foundation Grant 0600275 (0810902)
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.