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.

我们在具有边界的紧黎曼流形\((M^n,g)\)上引入了 Loewner-Nirenberg 问题的两种流动方法，并建立了相应的 Cauchy-Dirichlet 问题对 Loewner-Nirenberg 问题解的收敛性。特别是，当初始数据\(u_0\)是（1.1）的子解时，收敛性适用于直接流（1.3）-（1.5）和 Yamabe 流（1.6）。此外，当背景度量满足\(R_g\ge 0\) 时，对于直接流的任何正初始数据\(u_0\in C^{2,\alpha }(M)\)收敛成立；而对于这种情况，保形拉普拉斯算子\(L_g\)的狄利克雷问题的第一个特征值\(\lambda _1<0 \)，会聚保持用于\（U_0> V_0 \）其中\（V_0 \）是最大的溶液，以（1.1）均匀狄利克雷边界值问题和\（V_0> 0 \）在\（M ^ {\ CIRC} \)。我们还给出了在 ( M ,  g ) 和\(\inf _{u\in C^1(M)-\{0\}}Q的共形类中存在正标量曲率的度量之间的等效描述(u)>-\infty \)当 ( M ,  g ) 是光滑的，前提是正质量定理成立，其中Q是第二类 Escobar-Yamabe 问题的能量泛函（见（3.2））。