The $\sigma_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k<\frac{n}{2}$

Abstract

Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ with non-empty boundary $\partial M$. Let $\Gamma\subset\mathbb{R}^n$ be a symmetric convex cone and $f$ a symmetric defining function for $\Gamma$ satisfying standard assumptions. Under an algebraic condition on $\Gamma$, which is satisfied for example by the G\r{a}rding cones $\Gamma_k^+$ when $k<\frac{n}{2}$, we prove the existence of a Lipschitz viscosity solution $g_u = e^{2u}g_0$ to the fully nonlinear Loewner-Nirenberg problem associated to $(f,\Gamma)$, \begin{align*} \begin{cases} f(\lambda(-g_u^{-1}A_{g_u})) = 1, \quad \lambda(-g_u^{-1}A_{g_u}) \in \Gamma & \mathrm{on~}M\backslash\partial M \newline u(x)\rightarrow+\infty & \mathrm{as~}\operatorname{dist}_{g_0}(x,\partial M)\rightarrow 0, \end{cases} \end{align*} where $A_{g_u}$ is the Schouten tensor of $g_u$. Previous results on Euclidean domains show that, in general, $u$ is not differentiable. The solution $u$ is obtained as the limit of smooth solutions to a sequence of fully nonlinear Loewner-Nirenberg problems on approximating cones containing $(1,0,\dots,0)$, for which we also have uniqueness. In the process, we obtain an existence and uniqueness result for the corresponding Dirichlet boundary value problem with finite boundary data, which is also of independent interest. An important feature of our paper is that the existence of a conformal metric $g$ satisfying $\lambda(-g^{-1}A_g)\in\Gamma$ on $M$ is a consequence of our results, rather than an assumption.