The geometry of properly embedded special surfaces in R3; e.g., surfaces satisfying aH+bK=1, where a and b are positive

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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“…Moreover, if S is noncompact, then, from (16), (17), (19) and (20), its Gaussian curvature K I = K − 1 can be calculated as (23)…”

Section: Completeness Of the Immersionsmentioning

confidence: 99%

“…We call them linear Weingarten surfaces of Bryant type, in short, BLW-surfaces. These surfaces are a particular case of the special surfaces studied by Sa Earp and Toubiana in [21] (see also [17], [19] and [20]). …”

Section: Introductionmentioning

confidence: 99%

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

Gálvez

Martínez

Milán

2004

Trans. Amer. Math. Soc.

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“…As conclusion of Lemmas 6.1 and 6.2, we describe the behavior of the coordinates functions of the profile curve α under the assumption (12). See Figure 4 (b).…”

Section: Lemma 62mentioning

confidence: 86%

“…We only have to show that z(T ) = z 0 . But this direct from (13), the assumption (12) and that a/2 < −2b/a.…”

Section: Lemma 62mentioning

confidence: 97%

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Stationary surfaces in Lorentz–Minkowski space

López

2008

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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A linear Weingarten surface in Euclidean space R 3 is a surface whose mean curvature H and Gaussian curvature K satisfy a relation of the form aH + bK = c, where a, b, c ∈ R. Such a surface is said to be hyperbolic when a 2 + 4bc < 0. In this paper we study rotational linear Weingarten surfaces of hyperbolic type giving a classification under suitable hypothesis. As a consequence, we obtain a family of complete hyperbolic linear Weingarten surfaces in R 3 that consists of surfaces with self-intersections whose generating curves are periodic.

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Classification of rotational surfaces in Euclidean space satisfying a linear relation between their principal curvatures

López

Pámpano

2020

Mathematische Nachrichten

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We classify all rotational surfaces in Euclidean space whose principal curvatures κ1 and κ2 satisfy the linear relation κ1=aκ2+b, where a and b are two constants. As a consequence of this classification, we find closed (embedded and not embedded) surfaces and periodic (embedded and not embedded) surfaces with a geometric behaviour similar to Delaunay surfaces. Finally, we give a variational characterization of the generating curves of these surfaces.

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The geometry of properly embedded special surfaces in R3; e.g., surfaces satisfying aH+bK=1, where a and b are positive

Cited by 51 publications

References 6 publications

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

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

Stationary surfaces in Lorentz–Minkowski space

Classification of rotational surfaces in Euclidean space satisfying a linear relation between their principal curvatures

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