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 $$R_g\ge 0$$ , 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 $$M^{\circ }$$ . We also give an equivalent description between the existence of a metric of positive scalar curvature in the conformal class of (M, g) and $$\inf _{u\in C^1(M)-\{0\}}Q(u)>-\infty $$ 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.