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Prescribing Curvatures on Three Dimensional Riemannian Manifolds with Boundaries

Lei Zhang
Transactions of the American Mathematical Society
Vol. 361, No. 7 (Jul., 2009), pp. 3463-3481
Stable URL: http://www.jstor.org/stable/40302908
Page Count: 19
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Prescribing Curvatures on Three Dimensional Riemannian Manifolds with Boundaries
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Abstract

Let (M, g) be a complete three dimensional Riemannian manifold with boundary ∂M. Given smooth functions K(x) > 0 and c(x) defined on M and ∂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 M̄ some of them may appear on dM. 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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