Corpus ID: 16100276

# Surfaces of constant curvature in 3-dimensional space forms .

```@inproceedings{Glvez2009SurfacesOC,
title={Surfaces of constant curvature in 3-dimensional space forms .},
author={J. A. G{\'a}lvez},
year={2009}
}```
The study of surfaces immersed in a 3-dimensional ambient space plays a central role in the theory of submanifolds. In addition, Riemannian manifolds with constant sectional curvature can be considered as the most simple examples. Thus, one can think of surfaces with constant Gauss curvature in the Euclidean space R, hyperbolic space H or 3-sphere S as very natural objects of study. Through these notes we will study some classical results on complete surfaces with constant Gauss curvature in 3… Expand
12 Citations

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#### References

SHOWING 1-10 OF 45 REFERENCES
Complete surfaces with positive extrinsic curvature in product spaces
• Mathematics
• 2007
We prove that every complete connected immersed surface with positive extrinsic curvature K in H 2 � R must be properly embedded, homeomorphic to a sphere or a plane and, in the latter case, studyExpand
SOME USES OF THE SECOND CONFORMAL STRUCTURE ON STRICTLY CONVEX SURFACES
1. An oriented surface 5 immersed smoothly in E3 has a conformai structure imposed upon it by the metric of the surrounding space. Thus S may be viewed as a Riemann surface Pi. But if S is strictlyExpand
The Existence of Hypersurfaces of Constant Gauss Curvature with Prescribed Boundary
• Mathematics
• 2002
In this paper we are concerned with the problem of finding hypersurfaces of constant Gauss-Kronecker curvature (K-hypersurfaces) in R (n ≥ 2) with prescribed boundary: given a disjoint collection Γ =Expand
Flat surfaces in hyperbolic space as normal surfaces to a congruence of geodesics
We present some results concerning flat surfaces in hyperbolic 3-space, including: • A geometrical derivation for a local Weierstrass representation for flat surfaces, using as a starting point anExpand
Un lemme de Morse pour les surfaces convexes
We describe the "hyperbolic" properties of a riemann surface lamination M canonically associated to every compact three manifolds of curvature less than 1. More precisely, if the geodesic flow is theExpand
A rigidity theorem for the Clifford tori in
• Mathematics
• 1996
Let S3 be the unit hypersphere in the 4-dimensional Euclidean space R4 defined by ∑4 i=1 xi = 1. For each θ with 0 < θ < π/2, we denote by Mθ the Clifford torus in S 3 given by the equations x1 + x 2Expand
Embedded isolated singularities of flat surfaces in hyperbolic 3-space
• Mathematics
• 2005
Abstract.We give a complete description of the flat surfaces in hyperbolic 3-space that are regularly embedded around an isolated singularity. Specifically, we show that there is a one-to-oneExpand
Singularities of flat fronts in hyperbolic space
• Mathematics
• 2004
It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a front if it is the projection of aExpand
A Hopf differential for constant mean curvature surfaces inS2×R andH2×R
• Mathematics
• 2004
A basic tool in the theory of constant mean curvature (cmc) surfaces Σ in space forms is the holomorphic quadratic differential discovered by H. Hopf. In this paper we generalize this differential toExpand
Flat fronts in hyperbolic 3-space and their caustics
• Mathematics
• 2005
After Galvez, Martinez and Milan discovered a (Weierstrass-type) holomorphic representation formula for flat surfaces in hyperbolic 3-space, the first, third and fourth authors here gave a frameworkExpand