A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary.II

Abstract

This is a continuation of the research in [16]. Let $(\overline{M},g_{-1})$ be a closed geodesic $r_0$-ball in the hyperbolic space $(\mathbb{H}^n,g_{-1})$. Let $m\neq1$ be a positive constant. In this paper, we show that for $n\geq3$, starting from the metric $m g_{-1}$ on $\overline{M}$, with certain prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class $[g_{\mathbb{S}^{n-1}}]$ on the boundary $\partial M$, the solution $g(t)$ to the normalized Ricci flow $(1.2)$ which is continuous up to the boundary, exists for all $t>0$, and converges locally uniformly in the interior $M$ of $\overline{M}$ to a complete hyperbolic metric as $t\to\infty$(see Theorem 1.1 for details). Under some additional conditions, we show the same conclusion holds for $n=2$.