Normalized Ricci flow on geodesic balls in hyperbolic space with non-decreasing rotationally symmetric boundary mean curvature exists for all time and converges to a hyperbolic metric.
Pulemotov,Quasilinear parabolic equations and the Ricci flow on manifolds with boundary, J
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A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary.II
Normalized Ricci flow on geodesic balls in hyperbolic space with non-decreasing rotationally symmetric boundary mean curvature exists for all time and converges to a hyperbolic metric.