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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.

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Notes

  1. Indeed, by Corollary 1.7 below, \(u_\Omega \) is bounded on \({\mathbb {R}}^n\) and goes to zero at infinity. Since \(\Gamma \subset \Gamma _1\), \(u_\Omega \) is subharmonic in \({\mathbb {R}}^n {\setminus } \{0\}\) and thus in \({\mathbb {R}}^n\) as \(u_\Omega \) is bounded from below. This implies by the maximum principle that \(u_\Omega \le 0\) and so \(u_\Omega \equiv 0\).

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Acknowledgements

M.M. González is supported by Spanish government Grants MTM2014-52402-C3-1-P and MTM2017-85757-P, and the BBVA Foundation Grant for Researchers and Cultural Creators 2016. Y.Y. Li is partially supported by NSF Grant DMS-1501004. The first and second authors are grateful to the Fields Institute in Toronto for hospitality during part of work on this paper.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049, Madrid, Spain

    María del Mar González

  2. Department of Mathematics, Hill Center, Busch Campus, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854, USA

    YanYan Li

  3. Mathematical Institute and St Edmund Hall, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    Luc Nguyen

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  1. María del Mar González
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Correspondence to YanYan Li.

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González, M.d.M., Li, Y. & Nguyen, L. Existence and Uniqueness to a Fully Nonlinear Version of the Loewner–Nirenberg Problem. Commun. Math. Stat. 6, 269–288 (2018). https://doi.org/10.1007/s40304-018-0150-0

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