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

We study the regularity of viscosity solution u $u$ σ k $\sigma _k$ -Loewner–Nirenberg problem on a bounded smooth domain Ω ⊂ R n $\Omega \subset \mathbb {R}^n$ for ⩾ 2 $k \geqslant 2$ . It was known that is locally Lipschitz in $\Omega$ prove that, with d $d$ being distance function to ∂ $\partial \Omega$ and δ > 0 $\delta 0$ sufficiently small, { < ( x ) } $\lbrace d(x) \delta \rbrace$ first − 1 $(n-1)$ derivatives $d^{\frac{n-2}{2}} u$ are Hölder continuous ⩽ \leqslant Moreover, we identify boundary invariant which polynomial principal curvatures its covariant vanishes if only Using relation between Schouten tensor ambient manifold mean curvature submanifold related tools from geometric measure theory, further when contains more than one connected components, not differentiable