2025/02/02 by Li, Gang
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.00660
Let (M,g0) be a 2-D compact surface with boundary ∂ 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 ∂ 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.