A nonzero-degree map into a parabolically convex hyperbolic domain, with scalar curvature and boundary curvature bounds, forces the domain to be hyperbolic and the boundary map to be an isometry.
Total curvatur es of convex hypersurfaces in hy- perbolic space
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Scalar curvature rigidity of parabolically convex domains in hyperbolic spaces
A nonzero-degree map into a parabolically convex hyperbolic domain, with scalar curvature and boundary curvature bounds, forces the domain to be hyperbolic and the boundary map to be an isometry.