On the geometry of constant mean curvature one surfaces in hyperbolic space

Earp, Ricardo Sá; Toubiana, Éric

doi:10.1215/ijm/1258138346

Cited by 29 publications

(35 citation statements)

References 5 publications

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“…There is a certain amount of literature on CMC hypersurfaces in hyperbolic space [4,23,26,27]; but due to the non-compactness of hyperbolic space, this theory can be considered not such a vast departure from the theory of CMC hypersurfaces in R n+1 . Much less is known when the ambient space is the sphere.…”

Section: Gluing Constructions Of Cmc Hypersurfacesmentioning

confidence: 99%

Gluing constructions amongst constant mean curvature hypersurfaces in $${\mathbb {S}^{n+1}}$$

Butscher

2009

Ann Glob Anal Geom

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Four constructions of constant mean curvature (CMC) hypersurfaces in S n+1 are given, which should be considered analogues of 'classical' constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all the values of the mean curvature. Second, a hypersurface is constructed which consists of two chains of spheres winding around a pair of orthogonal equators, showing that Delaunay-like hypersurfaces can be fused together in a symmetric manner. Third, a Delaunay-like handle can be attached to a generalized Clifford torus of the same mean curvature. Finally, two generalized Clifford tori of equal but opposite mean curvature of any magnitude can be attached to each other by symmetrically positioned Delaunay-like 'arms'. This last result extends Butscher and Pacard's doubling construction for generalized Clifford tori of small mean curvature.

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Section: Gluing Constructions Of Cmc Hypersurfacesmentioning

confidence: 99%

Gluing constructions amongst constant mean curvature hypersurfaces in $${\mathbb {S}^{n+1}}$$

Butscher

2009

Ann Glob Anal Geom

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“…The corresponding characterization in the Euclidean case is that a minimal end is planar if and only if it is the real part of a holomorphic null immersion into C 3 with a simple pole at the end. In the case of Bryant surfaces there are two types of smooth ends: those asymptotic to horospheres and those asymptotic to smooth catenoid cousins in the sense of [25]. At a smooth horospherical end F has a simple pole and F ′ F −1 automatically has a second order pole.…”

Section: Introductionmentioning

confidence: 99%

Bryant surfaces with smooth ends

Bohle

Peters

2009

Communications in Analysis and Geometry

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A smooth end of a Bryant surface is a conformally immersed punctured disc of mean curvature 1 in hyperbolic space that extends smoothly through the ideal boundary. The Bryant representation of a smooth end is well defined on the punctured disc and has a pole at the puncture. The Willmore energy of compact Bryant surfaces with smooth ends is quantized. It equals 4π times the total pole order of its Bryant representation. The possible Willmore energies of Bryant spheres with smooth ends are 4π(N * \ {2, 3, 5, 7}). Bryant spheres with smooth ends are examples of soliton spheres, a class of rational conformal immersions of the sphere which also includes Willmore spheres in the conformal 3-sphere S 3 . We give explicit examples of Bryant spheres with an arbitrary number of smooth ends. We conclude the paper by showing that Bryant's quartic differential Q vanishes identically for a compact surface in S 3 if and only if it is the compactification of either a complete finite total curvature Euclidean minimal surface with planar ends or a compact Bryant surface with smooth ends.

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“…The norm of an element n ∈ sl(2, C) is just n, n /8 = det(n). Similar to the case of C 3 , null curves in Sl(2, C) are in close relation to surfaces of constant curvature in the hyperbolic space H 3 (see for instance [4,12] or [13]). If Ψ : U ⊂ C → Sl(2, C) is a non-constant null curve, then the MaurerCartan derivative n = Ψ −1 dΨ : U → sl(2, C) is a matrix valued 1-form n = n 11 n 12 n 21 n 22 with vanishing determinant and trace.…”

Section: Duals Of Derivatives Of Curves In Sl(2 C)mentioning

confidence: 84%

Duals of Vector Fields and of Null Curves

Gollek

2007

Result. Math.

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The notion of dual curve of projective differential geometry is generalized to a dualization operator on the space of curves in the tangent bundle of a real or complex Riemannian manifold M and on the space of vector fields too. In dimension 3 it turns out that dualization maps null curves to isotropic curves and vice versa. We study this dualization in the special cases of M = C 3 and M = Sl(2, C) and give analytic descriptions in terms of the Weierstraß-, respectively Bryant data, of null curves in M = C 3 resp. M = Sl(2, C). Mathematics Subject Classification (2000). Primary 53A10; Secondary 53B20, 53A20.

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On the geometry of constant mean curvature one surfaces in hyperbolic space

Cited by 29 publications

References 5 publications

Gluing constructions amongst constant mean curvature hypersurfaces in $${\mathbb {S}^{n+1}}$$

Gluing constructions amongst constant mean curvature hypersurfaces in $${\mathbb {S}^{n+1}}$$

Bryant surfaces with smooth ends

Duals of Vector Fields and of Null Curves

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