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

TL;DR

This paper solves a class of fully nonlinear conformal geometry problems on compact manifolds with boundary, extending previous results to include the critical case where the order equals half the dimension, and establishing existence under certain geometric conditions.

Contribution

The authors extend the solvability of the $\sigma_k$-Loewner-Nirenberg problem to the case $k=rac{n}{2}$, including new existence results without restrictions on $ ext{mu}_ ext{Gamma}^+$.

Findings

Existence of solutions when $ ext{mu}_ ext{Gamma}^+ > 1- ext{delta}$.

Solutions to the boundary value problem with positive boundary data under certain conformal metric conditions.

Extension of previous work to include the critical threshold case $k=rac{n}{2}$.

Abstract

Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ 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(\lambda(-g_u^{-1}A_{g_u})) = \frac{1}{2}, \quad \lambda(-g_u^{-1}A_{g_u})\in\Gamma & \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 $k\leq \frac{n}{2}$, 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 $(1,0,\dots,0)\in\partial\Gamma$.