A normalized Ricci flow on surfaces with boundary towards the complete hyperbolic metric
Gang Li
TL;DR
This work analyzes the normalized Ricci flow on a compact 2D surface with boundary, recasting the flow in conformal form as $u_t=e^{-2u}\Delta_g u-1$ with Robin boundary data $\frac{\partial u}{\partial n_g}+k_g=\psi e^u$, and seeks convergence to the Loewner-Nirenberg complete hyperbolic metric. It introduces an auxiliary Cauchy-Dirichlet problem to obtain crucial a priori estimates, proves interior upper bounds and a comparison principle, and establishes long-time existence and convergence to $u_{LN}$ under carefully controlled boundary data $\phi(t)$ that determine the asymptotic boundary geodesic curvature. The paper also derives detailed boundary-behavior results: slow boundary-data growth yields $k_{e^{2u}g}\to1$, while fast growth produces $k_{e^{2u}g}\to\infty$, and it constructs barrier arguments to show convergence for a broad class of initial and boundary data. It further discusses extensions to higher dimensions and poses open problems on intrinsic boundary prescriptions and conformal flows with prescribed boundary curvature.
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 $ψ$ 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 $ψ>0$ 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 $ψ$ 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.