A normalized Ricci flow on surfaces with boundary towards the complete hyperbolic metric

TL;DR

This paper proves that the normalized Ricci flow on surfaces with boundary, under certain conditions, converges to a complete hyperbolic metric, with detailed analysis of boundary curvature effects and convergence criteria.

Contribution

It establishes existence, uniqueness, and convergence of the normalized Ricci flow on surfaces with boundary towards the hyperbolic metric, including handling boundary geodesic curvature conditions.

Findings

Flow converges to complete hyperbolic metric under positive boundary curvature

Existence and uniqueness of solutions for all positive time

Examples showing boundary data can prevent convergence

Abstract

Let $(\overline{M},g_0)$ be a $2$-D compact surface with boundary $\partial M$ and its interior $M$. We show that for a large class of initial and boundary data, the initial-boundary value problem of the normalized Ricci flow $(1.10)-(1.12)$, with prescribed geodesic curvature $\psi$ on $\partial M$, has a unique solution for all $t>0$, and it converges to the complete hyperbolic metric locally uniformly in $M$. Here the natural condition that $\psi>0$ causes the main difficulty in the a priori estimates in the corresponding initial-boundary problem $(1.15)-(1.17)$ of the parabolic equations, for which an auxiliary Cauchy-Dirichlet problem is introduced. We also provide examples of the boundary data $\psi$ which fits well with the natural asymptotic behavior of the geodesic curvature, but the solution to $(1.10)-(1.12)$ fails to converge to the complete hyperbolic metric.