Two flow approaches to the Loewner-Nirenberg problem on manifolds

TL;DR

This paper develops two flow methods to solve the Loewner--Nirenberg problem on compact Riemannian manifolds with boundary, proving convergence under various initial conditions and relating solutions to scalar curvature metrics.

Contribution

Introduces and analyzes two flow approaches to the Loewner--Nirenberg problem, establishing convergence criteria and linking solutions to scalar curvature conditions.

Findings

Convergence of flow solutions to the Loewner--Nirenberg problem under specific initial data.

Conditions on the background metric affect convergence of the flows.

Characterization of positive scalar curvature metrics via energy functional bounds.

Abstract

We introduce two flow approaches to the Loewner--Nirenberg problem on comapct Riemannian manifolds $(M^n,g)$ with boundary and establish the convergence of the corresponding Cauchy--Dirichlet problems to the solution of the Loewner--Nirenberg problem. In particular, when the initial data $u_0\in C^{4,\alpha}(M)$ is a solution or a strict subsolution to the equation $(1.1)$, the convergence holds for both the direct flow $(1.3)-(1.5)$ and the Yamabe flow $(1.10)$. Moreover, when the background metric satisfies $R_g\geq0$, the convergence holds for any positive initial data $u_0\in C^{2,\alpha}(M)$ for the direct flow; while for the case the first eigenvalue $\lambda_1<0$ for the Dirichlet problem of the conformal Laplacian $L_g$, the convergence holds for $u_0>v_0$ where $v_0$ is the largest solution to the homogeneous Dirichlet boundary value problem of $(1.1)$ and $v_0>0$ in $M^{\circ}$ (the interior of $M$). We also give an equivalent description between the existence of a metric of positive scalar curvature in the conformal class of $(M,g)$ and $\displaystyle\inf_{u\in C^1(M),\,u\not\equiv 0\,\text{on}\,\partial M}Q(u)>-\infty$, where $Q$ is the energy functional (see $(1.8)$) of the second type Escobar-Yamabe problem.