Normalized Ricci flow on a geodesic ball in hyperbolic space with prescribed boundary mean curvature exists forever and converges to a complete hyperbolic metric.
Schn ¨urer, F
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A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary
Normalized Ricci flow on a geodesic ball in hyperbolic space with prescribed boundary mean curvature exists forever and converges to a complete hyperbolic metric.