Rotational surfaces of constant astigmatism in space forms

López, Rafael; Pámpano, Álvaro

doi:10.1016/j.jmaa.2019.123602

Cited by 5 publications

(5 citation statements)

References 19 publications

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“…(i) c < 1: Then a 2 = µ 1−c is well defined, and putting b 2 = a 4 µ , we conclude that (25) is exactly (15) and Corollary 2 gives that we arrive at the ellipsoid of revolution (13). In particular, if c = 0, we obtain the sphere of radius √ µ.…”

Section: Cubic Rotational Weingarten Surfacesmentioning

confidence: 87%

“…For our purposes, recalling Corollary 2, we need to compute the geometric linear momentum of the ellipsoid (13). We parametrize the generatrix semiellipse by x = a cos t, z = b sin t, t ∈ [−π/2, π/2], and using Remark 1, it is not difficult to conclude that…”

Section: Cubic Rotational Weingarten Surfacesmentioning

confidence: 99%

“…However, the complete classification of rotational linear Weingarten surfaces has not been achieved surprisingly until 2020 in [12]. Some other interesting rotational Weingarten surfaces have been also recently studied in [13,14]. We refer to [8] (and references therein) for the study of closed rotational Weingarten surfaces satisfying κ 1 = c κ 2n+1 2 , generalizing results of Hopf [2] when n = 0 and the case of ellipsoids of revolution when n = 1.…”

Section: Introductionmentioning

confidence: 95%

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A New Approach to Rotational Weingarten Surfaces

Carretero

Castro

2022

Mathematics

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Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first-order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an hyperbola in the W-diagram of the surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids, defined as the rotational surfaces generated by the rotation of the Euler elastic curves around their directrix line. As another application of our approach, we deal with the problem of prescribing mean or Gauss curvature on rotational surfaces in terms of arbitrary continuous functions depending on distance from the surface to the axis of revolution. As a consequence, we provide simple new proofs of some classical results concerning rotational surfaces, such as Euler’s theorem about minimal ones, Delaunay’s theorem on constant mean curvature ones, and Darboux’s theorem about constant Gauss curvature ones.

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Section: Cubic Rotational Weingarten Surfacesmentioning

confidence: 87%

Section: Cubic Rotational Weingarten Surfacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

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A New Approach to Rotational Weingarten Surfaces

Carretero

Castro

2022

Mathematics

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“…However, the complete classification of rotational linear Weingarten surfaces has not been achieved surprisingly until 2020 in [LP20a]. Some other interesting rotational Weingarten surfaces have been also recently studied in [LP20b] and [LP20c]. We refer to [KS05] (and references therein) for the study of closed rotational Weingarten surfaces satisfying κ 1 = c κ 2n+1 2 , generalizing results of Hopf [H51] when n = 0 and the case of ellipsoids of revolution when n = 1.…”

Section: Introductionmentioning

confidence: 93%

A new approach to rotational Weingarten surfaces

Carretero¹,

Castro²

2021

Preprint

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Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an hyperbola in the Wdiagram of the surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids, defined as the rotational surfaces generated by the rotation of the Euler elastic curves around their directrix line.As another application of our approach, we deal with the problem of prescribing mean or Gauss curvature on rotational surfaces in terms of arbitrary continuous functions depending on distance from the surface to the axis of revolution. As a consequence, we provide simple new proofs of some classical results concerning rotational surfaces, like Euler's theorem about minimal ones, Delaunay's theorem on constant mean curvature ones, and Darboux's theorem about constant Gauss curvature ones.

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“…This energy was introduced in [24] in order to study rotational surfaces of constant astigmatism in Riemannian 3-space forms. Moreover, by studying some geometric properties of critical curves and the closure condition, in the same paper it was proved that the only closed critical curves with non-constant curvature appear in S 2 (ρ).…”

Section: Examples In Homogeneous 3-spacesmentioning

confidence: 99%

Critical tori for mean curvature energies in Killing submersions

Pámpano

2020

Nonlinear Analysis

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We study surface energies depending on the mean curvature in total spaces of Killing submersions, which extend the classical notion of Willmore energy. Based on a symmetry reduction procedure, we construct vertical tori critical for these mean curvature energies. These vertical tori are based on closed curves critical for curvature energy functionals in Riemannian 2-space forms.The binormal evolution of these critical curves in Riemannian 3-space forms generates rotational tori solutions for an ample family of Weingarten surfaces. Therefore, we also introduce some correspondence results between these two types of tori and illustrate their relation.

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Rotational surfaces of constant astigmatism in space forms

Cited by 5 publications

References 19 publications

A New Approach to Rotational Weingarten Surfaces

A New Approach to Rotational Weingarten Surfaces

A new approach to rotational Weingarten surfaces

Critical tori for mean curvature energies in Killing submersions

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