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

TL;DR

This paper studies the long-term behavior of the normalized Ricci flow on certain compact manifolds with boundary, showing convergence to hyperbolic metrics under specific conditions, extending previous results to the case n=2.

Contribution

It demonstrates the existence, convergence, and boundary behavior of solutions to the Ricci flow on manifolds with boundary, including the case n=2, with prescribed boundary conditions.

Findings

Solutions exist for all positive time and converge to hyperbolic metrics.

Convergence holds under prescribed boundary mean curvature and conformal class.

Results extend to the case n=2 with additional conditions.

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 $n=2$.