Existence and Uniqueness to a Fully Nonlinear Version of the Loewner–Nirenberg Problem

María del Mar González , YanYan Li , Luc Nguyen

Communications in Mathematics and Statistics ›› 2018, Vol. 6 ›› Issue (3) : 269 -288.

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Communications in Mathematics and Statistics ›› 2018, Vol. 6 ›› Issue (3) : 269 -288. DOI: 10.1007/s40304-018-0150-0
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Existence and Uniqueness to a Fully Nonlinear Version of the Loewner–Nirenberg Problem

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Abstract

We consider the problem of finding on a given Euclidean domain $\Omega $ of dimension $n \ge 3$ a complete conformally flat metric whose Schouten curvature A satisfies some equations of the form $f(\lambda (-A)) = 1$. This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of such metric when the boundary $\partial \Omega $ is a smooth bounded hypersurface (of codimension one). When $\partial \Omega $ contains a compact smooth submanifold $\Sigma $ of higher codimension with $\partial \Omega {\setminus }\Sigma $ being compact, we also give a ‘sharp’ condition for the divergence to infinity of the conformal factor near $\Sigma $ in terms of the codimension.

Keywords

Fully nonlinear Loewner–Nirenberg problem / Singular fully nonlinear Yamabe metrics / Conformal invariance

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María del Mar González, YanYan Li, Luc Nguyen. Existence and Uniqueness to a Fully Nonlinear Version of the Loewner–Nirenberg Problem. Communications in Mathematics and Statistics, 2018, 6(3): 269-288 DOI:10.1007/s40304-018-0150-0

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