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

TL;DR

This paper demonstrates that under certain conditions, the normalized Ricci flow on a geodesic ball in hyperbolic space converges to a complete hyperbolic metric, with sectional curvature remaining below -1 for all positive times.

Contribution

It establishes long-time existence and convergence of the Ricci flow on compact manifolds with boundary to hyperbolic metrics, extending previous results with new boundary conditions.

Findings

Flow converges to a complete hyperbolic metric as time approaches infinity.

Sectional curvature remains less than -1 for all positive times.

Additional boundary curvature conditions are needed in dimension 2 for convergence.

Abstract

In this paper, we show that starting from a geodesic ball $\overline{B_{r_0}}(0)$ in $\mathbb{H}^n$, for $n\geq3$, with prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class $[g_{\mathbb{S}^{n-1}}]$ on the boundary, 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 $B_{r_0}(0)$ to a complete hyperbolic metric as $t\to\infty$(see Theorem 1.2 for details). Moreover, the sectional curvature of $g(t)$ maintains less than $-1$ for $t>0$. For dimension $2$, to achieve such a convergence result, we need the additional assumption that the mean curvature on the boundary increases in a certain speed to infinity as $t\to\infty$.