The ${\sigma }_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k=\frac{n}{2}$ and beyond

Abstract

Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\ge 3$ with smooth 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. Denoting by $A_{g_u}$ the Schouten tensor of a conformal metric $g_u=u^{-2}{g}_0$, we show that the associated fully nonlinear Loewner-Nirenberg problem \begin{align*} \begin{cases} f(λ(-g_u^{-1}A_{g_u})) = \frac{1}{2}, \quad λ(-g_u^{-1}A_{g_u})\inΓ& \text{on }M\backslash \partial M \newline u = 0 & \text{on }\partial M \end{cases} \end{align*} admits a solution if ${\mu }_{\Gamma }^{+}>1-\delta$, where ${\mu }_{\Gamma }^{+}$ is defined by $(-{\mu }_{\Gamma }^{+},1,\dots ,1)\in \partial \Gamma$ and $\delta >0$ is a constant depending on certain geometric data. In particular, we solve the ${\sigma }_k$-Loewner-Nirenberg problem for all $k\le \frac{n}{2}$, which extends recent work of the authors to include the important threshold case $k=\frac{n}{2}$. In the process, we establish that the fully nonlinear Loewner-Nirenberg problem and corresponding Dirichlet boundary value problem with positive boundary data admit solutions if there exists a conformal metric $g\in [g_0]$ such that $\lambda (-g^{-1}{A}_g)\in \Gamma$ on $M$; these latter results require no assumption on ${\mu }_{\Gamma }^{+}$ and are new when $(1,0,\dots ,0)\in \partial \Gamma$.