# Hyperbolization of cusps with convex boundary

@article{Fillastre2015HyperbolizationOC, title={Hyperbolization of cusps with convex boundary}, author={Franccois Fillastre and Ivan Izmestiev and Giona Veronelli}, journal={Manuscripta Mathematica}, year={2015}, volume={150}, pages={475-492} }

We prove that for every metric on the torus with curvature bounded from below by −1 in the sense of Alexandrov there exists a hyperbolic cusp with convex boundary such that the induced metric on the boundary is the given metric. The proof is by polyhedral approximation. This was the last open case of a general theorem: every metric with curvature bounded from below on a compact surface is isometric to a convex surface in a 3-dimensional space form.

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#### References

SHOWING 1-10 OF 46 REFERENCES

Hyperbolic cusps with convex polyhedral boundary

- Mathematics
- 2007

We prove that a 3-dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthemore, any hyperbolic metric on the torus with cone… Expand

Hyperbolic manifolds with convex boundary

- Mathematics
- 2006

Let (M,∂M) be a 3-manifold, which carries a hyperbolic metric with convex boundary. We consider the hyperbolic metrics on M such that the boundary is smooth and strictly convex. We show that the… Expand

Polyhedral metrics on the boundaries of convex compact quasi-Fuchsian manifolds

- Mathematics
- 2014

Abstract We show the existence of a convex compact subset in a quasi-Fuchsian manifold such that the induced metric on the boundary of the subset coincides with a prescribed hyperbolic polyhedral… Expand

The Weyl Problem With Nonnegative Gauss Curvature In Hyperbolic Space

- Mathematics
- Canadian Journal of Mathematics
- 2015

Abstract In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form $H$… Expand

Compact domains with prescribed convex boundary metrics in quasi-Fuchsian manifolds

- Mathematics
- 2014

We show the existence of a convex compact domain in a quasi-Fuchsian manifold such that the induced metric on its boundary coincides with a prescribed surface metric of curvature $K\geq-1$ in the… Expand

A Variational Proof of Alexandrov’s Convex Cap Theorem

- Mathematics, Computer Science
- Discret. Comput. Geom.
- 2008

The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps to prove that generalized conveX caps with the fixed metric on the boundary are globally rigid, that is uniquely determined by their curvatures. Expand

Hypersurfaces of Constant Curvature in Hyperbolic Space I

- Mathematics
- 2008

We investigate the problem of finding, in hyperbolic space, a complete strictly convex hypersurface which has a prescribed asymptotic boundary at infinity and which has some fixed curvature function… Expand

Moderate smoothness of most Alexandrov surfaces

- Mathematics
- 2013

We show that, in the sense of Baire categories, a typical Alexandrov surface with curvature bounded below by κ has no conical points. We use this result to prove that, on such a surface (unless it is… Expand

ALEXANDROV'S THEOREM, WEIGHTED DELAUNAY TRIANGULATIONS, AND MIXED VOLUMES

- Mathematics
- 2006

We present a constructive proof of Alexandrov’s theorem regarding the existence of a convex polytope with a given metric on the boundary. The polytope is obtained as a result of a certain deformation… Expand

An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds

- Mathematics
- 2008

It is proved that a convex hypersurface in a Riemannian manifold of sectional curvature > κ is an Alexandrov’s space of curvature > κ . This theorem provides an optimal lower curvature bound for an… Expand