Existence and uniqueness to a fully non-linear version of the Loewner-Nirenberg problem

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 equation 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.

References in corpus (7)

• Fully nonlinear equations on Riemannian manifolds with negative curvature
• Asymptotic behavior of solutions to the ${\sigma }_k$-Yamabe equation near isolated singularities
• A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound
• Renormalized Volume
• Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations
• Harnack inequalities and Bôcher-type theorems for conformally invariant fully nonlinear degenerate elliptic equations
• Thinness for Scalar-Negative Singular Yamabe Metrics

Cited by in corpus (4)

• Regularity of viscosity solutions of the ${\sigma }_k$-Loewner-Nirenberg problem
• Local pointwise second derivative estimates for strong solutions to the ${\sigma }_k$-Yamabe equation on Euclidean domains
• The ${\sigma }_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k<\frac{n}{2}$
• Comparison principles for some fully nonlinear subelliptic equations on the Heisenberg group