Two Flow Approaches to the Loewner–Nirenberg Problem on Manifolds

Abstract

We introduce two flow approaches to the Loewner-Nirenberg problem on compact 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 is a subsolution to (1.1), the convergence holds for both the direct flow (1.3)-(1.5) and the Yamabe flow (1.6). Moreover, when the background metric satisfies R g ≥ 0, the convergence holds for any positive initial data u 0 ∈ C 2,α (M) for the direct flow; while for the case the first eigenvalue λ 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 • . We also give an equivalent description between the existence of a metric of positive scalar curvature in the conformal class of (M, g) and inf u∈C 1 (M)−{0} Q(u) > −∞ when (M, g) is smooth, provided that the positive mass theorem holds, where Q is the energy functional (see (3.2)) of the second type Escobar-Yamabe problem.

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

“…Let (M n , g) be a compact Riemannian manifold with boundary of dimension n ≥ 3. Denote M • to be the interior of M. In [10], we considered the Cauchy-Dirichlet problem of the Yamabe flow which starts from a positive subsolution of the Yamabe equation (1.1) and converges in C 2 loc (M • ) to the solution to the Loewner-Nirenberg problem 4(n − 1)…”