Isometric immersions of a Euclidean plane in Lobachevskii space

Volkov, Yu. A.; Vladimirova, S. M.

doi:10.1007/bf01464724

Cited by 14 publications

(20 citation statements)

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“…Corollary 3.5 ([20], [21]). Any complete flat surface in H 3 must be congruent to a horosphere or a hyperbolic cylinder.…”

Section: Applicationsmentioning

confidence: 99%

A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

Kawakami

2013

Geom Dedicata

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Abstract. We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. Moreover we give the two applications of this theorem, the first one is to show an analogue of the Ahlfors islands theorem for it and the second one is to give a simple proof of the classification of complete nonsingular flat surfaces in the hyperbolic three-space.

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“…Corollary 3.5 ([20], [21]). Any complete flat surface in H 3 must be congruent to a horosphere or a hyperbolic cylinder.…”

Section: Applicationsmentioning

confidence: 99%

A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

Kawakami

2013

Geom Dedicata

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“…If Σ is not simply connected, then we consider that ρ is a meromorphic function on its universal covering surface Σ. As a corollary of Theorem 3.8, we give a simple proof of the classification ( [35], [39]) of complete nonsingular flat surfaces in H 3 . For the proof, see [17,Corollary 3.5].…”

Section: 3mentioning

confidence: 99%

On the maximal number of exceptional values of Gauss maps for various classes of surfaces

Kawakami

2012

Math. Z.

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The main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of Gauss maps for several classes of immersed surfaces in space forms, for example, complete minimal surfaces in the Euclidean three-space, weakly complete improper affine spheres in the affine three-space and weakly complete flat surfaces in the hyperbolic three-space. For this purpose, we give an effective curvature bound for a specified conformal metric on an open Riemann surface.2010 Mathematics Subject Classification. Primary 30D35, 53C42 ; Secondary 30F45, 53A10, 53A15.

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“…Corollary 3.9 ( [49,52]). Any complete nonsingular flat surface in H 3 must be a horosphere or a hyperbolic cylinder.…”

Section: 2mentioning

confidence: 99%

Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces

Kawakami¹

2015

Can. j. math.

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Abstract. We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

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Isometric immersions of a Euclidean plane in Lobachevskii space

Cited by 14 publications

References 1 publication

A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

On the maximal number of exceptional values of Gauss maps for various classes of surfaces

Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces

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