A flow approach to the generalized Loewner-Nirenberg problem of the $\sigma_k$-Ricci equation

TL;DR

This paper develops a flow method to solve the generalized Loewner-Nirenberg problem for the $\sigma_k$-Ricci equation on compact manifolds with boundary, proving existence, uniqueness, and convergence of solutions.

Contribution

It introduces a flow-based approach to the problem, establishing convergence to the solution under subsolution initial data.

Findings

Existence and uniqueness of the flow solution.

Convergence of the flow to the boundary value problem solution.

Solution regularity in $C^4_{loc}$.

Abstract

We introduce a flow approach to the generalized Loewner-Nirenberg problem $(1.5)-(1.7)$ of the $\sigma_k$-Ricci equation on a compact manifold $(M^n,g)$ with boundary. We prove that for initial data $u_0\in C^{4,\alpha}(M)$ which is a subsolution to the $\sigma_k$-Ricci equation $(1.5)$, the Cauchy-Dirichlet problem $(3.1)-(3.3)$ has a unique solution $u$ which converges in $C^4_{loc}(M^{\circ})$ to the solution $u_{\infty}$ of the problem $(1.5)-(1.7)$, as $t\to\infty$.