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ISSN 1088-6850(e) ISSN 0002-9947(p)

Prescribing curvatures on three dimensional Riemannian manifolds with boundaries

Author(s): Lei Zhang
Journal: Trans. Amer. Math. Soc. 361 (2009), 3463-3481.
MSC (2000): Primary 35J60, 53B20
Posted: February 23, 2009
MathSciNet review: 2491888
Retrieve article in: PDF

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Abstract: Let be a complete three dimensional Riemannian manifold with boundary $\partial M$ . Given smooth functions and defined on and $\partial M$ , respectively, it is natural to ask whether there exist metrics conformal to so that under these new metrics, is the scalar curvature and is the boundary mean curvature. All such metrics can be described by a prescribing curvature equation with a boundary condition. With suitable assumptions on , and 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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