Regularity of viscosity solutions of the σk$\sigma _k$‐Loewner–Nirenberg problem

Abstract

AbstractWe study the regularity of the viscosity solution of the ‐Loewner–Nirenberg problem on a bounded smooth domain for . It was known that is locally Lipschitz in . We prove that, with being the distance function to and sufficiently small, is smooth in and the first derivatives of are Hölder continuous in . Moreover, we identify a boundary invariant which is a polynomial of the principal curvatures of and its covariant derivatives and vanishes if and only if is smooth in . Using a relation between the Schouten tensor of the ambient manifold and the mean curvature of a submanifold and related tools from geometric measure theory, we further prove that, when contains more than one connected components, is not differentiable in .