Complete Surfaces of Constant Mean Curvature-1 in the Hyperbolic 3-space

Umehara, Masaaki; Yamada, Kotaro

doi:10.2307/2946533

Cited by 120 publications

(194 citation statements)

References 5 publications

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“…In particular we have the following corollary. 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 The fact that θ = π 2 suggests that this correspondence looks like the conjugate cousin correspondence between minimal surfaces in R 3 and CMC 1 surfaces in H 3 ( [Bry87], [UY93]). This correspondence has nice geometric properties, and is useful to construct CMC 1 surfaces in H 3 with some prescribed geometric properties starting from a solution of a Plateau problem in R 3 (see for example [Kar05], [Dan06]).…”

Section: A Generalized Lawson Correspondencementioning

confidence: 99%

Isometric immersions into 3-dimensional homogeneous manifolds

Daniel¹

2007

Comment. Math. Helv.

180

296

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Abstract. We give a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous Riemannian manifold with a 4-dimensional isometry group. The condition is expressed in terms of the metric, the second fundamental form, and data arising from an ambient Killing field. This class of 3-manifolds includes in particular the Berger spheres, the Heisenberg group Nil 3 , the universal cover of the Lie group PSL 2 (R) and the product spaces S 2 × R and H 2 × R. We give some applications to constant mean curvature (CMC) surfaces in these manifolds; in particular we prove the existence of a generalized Lawson correspondence, i.e., a local isometric correspondence between CMC surfaces in homogeneous 3-manifolds. Mathematics Subject Classification (2000). Primary: 53C42. Secondary: 53A35, 53B25.

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Section: A Generalized Lawson Correspondencementioning

confidence: 99%

Isometric immersions into 3-dimensional homogeneous manifolds

Daniel¹

2007

Comment. Math. Helv.

180

296

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“…that is, the conformal representation becomes the Bryant's representation (see [2], [25]). Moreover, if the immersion does not lie in a horosphere and we denote by G its hyperbolic Gauss map, then taking A = G and B = 1 in (15) one gets from (21) and (22) …”

Section: Proof Let Us Assume That ψ(S)mentioning

confidence: 99%

“…Thus, the study of complete BLW-surfaces in R 4 can be reduced to the study of complete immersions with constant mean curvature one. Many things are known in this case and some very interesting results were proved in [2], [3] and [25].…”

Section: 2mentioning

confidence: 99%

“…Since Bryant's work many properties and examples have been discovered in [3], [15], [22], [25], [26] among others. By considering the complex structure determined by the second fundamental form, the authors proved in [6] that a "holomorphic resolution" like the Bryant representation also holds for flat surfaces in H 3 and used it in the study of some examples and the behaviour at infinity of complete ends.…”

Section: Introductionmentioning

confidence: 99%

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Complete linear Weingarten surfaces of Bryant type. A Plateau problem at infinity

Gálvez

Martínez

Milán

2004

Trans. Amer. Math. Soc.

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Abstract. In this paper we study a large class of Weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3-space. We show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the Euclidean 3-space and we use it to study their completeness. We also establish some existence and uniqueness theorems by studing the Plateau problem at infinity: when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family? and, how many embedded solutions are there?

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“…(Umehara and Yamada, Rosenberg, and others have been studying embeddedness of CMC 1 surfaces in H 3 [CHR1], [ET1], [LR], [UY1].) The goals of this article are three-fold:…”

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confidence: 99%

Mean curvature one surfaces in hyperbolic space, and their relationship to minimal surfaces in Euclidean space

Rossman

2001

J Geom Anal

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Abstract. We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean curvature 1 surfaces in hyperbolic 3-space, and we give an overview of recent results on these surfaces. We include computer graphics of a number of examples.

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Complete Surfaces of Constant Mean Curvature-1 in the Hyperbolic 3-space

Cited by 120 publications

References 5 publications

Isometric immersions into 3-dimensional homogeneous manifolds

Isometric immersions into 3-dimensional homogeneous manifolds

Complete linear Weingarten surfaces of Bryant type. A Plateau problem at infinity

Mean curvature one surfaces in hyperbolic space, and their relationship to minimal surfaces in Euclidean space

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