Topological and metric rigidity teorems for hypersurfaces in a hyperbolic space

Topological Rigidity for Hyperbolic Manifolds

Let M be a complete Riemannian manifold having constant sectional curvature —1 and finite volume. Let M denote its Gromov-Margulis manifold compactification and assume that the dimension of M is greater than 5. (If M is compact, then M = M and dM is empty.) We announce (among other results) that any homotopy equivalence h: (N,dN) —• (M,dM), where N is a compact manifold, is homotopic to a homeo...

Rigidity Results for Hypersurfaces in Space Forms

Symmetry of stationary hypersurfaces in hyperbolic space

We deposit a prescribed amount of liquid on an umbilical hypersurface of the hyperbolic space Hn+1. Under the presence of a uniform gravity vector field directed towards , we seek the shape of such a liquid drop in a state of equilibrium of the mechanical system. The liquid-air interface is then modeled by a hypersurface under the condition that its mean curvature is a function of the distance ...

Rigidity and Stability for Isometry Groups in Hyperbolic 4-Space

Rigidity and Stability for Isometry Groups in Hyperbolic 4-Space by Youngju Kim Advisor: Professor Ara Basmajian It is known that a geometrically finite Kleinian group is quasiconformally stable. We prove that this quasiconformal stability cannot be generalized in 4-dimensional hyperbolic space. This is due to the presence of screw parabolic isometries in dimension 4. These isometries are topol...

Minkowski Type Problems for Convex Hypersurfaces in Hyperbolic Space

We consider the problem F = f(ν) for strictly convex, closed hypersurfaces in H and solve it for curvature functions F the inverses of which are of class (K).