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Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq3$ with non-empty boundary $\partial M$. Let $\Gamma\subset\mathbb{R}^n$ be asymmetric 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,\partialM)\rightarrow 0,  \end{cases}  \end{align*}  where $A_{g_u}$ is the Schouten tensor of $g_u$. Previous results onEuclidean domains show that, in general, $u$ is not differentiable. Thesolution $u$ is obtained as the limit of smooth solutions to a sequence offully nonlinear Loewner-Nirenberg problems on approximating cones containing$(1,0,\dots,0)$, for which we also have uniqueness. In the process, we obtainan existence and uniqueness result for the corresponding Dirichlet boundaryvalue 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 ourresults, rather than an assumption.

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