Willmore-Like Tori in Killing Submersions

Barros, Manuel; Garay, Óscar J.; Pámpano, Álvaro

doi:10.1155/2018/4652516

Cited by 3 publications

(5 citation statements)

References 30 publications

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“…Now, for a given function τ ∈ C ∞ (B(ρ)) we construct the Killing submersion over B(ρ) with bundle curvature τ , π : M (ρ, τ ) → B(ρ). Recall that the existence of these Killing submersions was proved in [10] (see also [25]). Finally, we apply Theorem 3.1 to draw the conclusion.…”

Section: Correspondence Resultsmentioning

confidence: 94%

“…A natural question that arises here concerns the existence of Killing submersions over a given surface B (with Gaussian curvature K B ) for a prescribed bundle curvature τ ∈ C ∞ (B). For arbitrary Riemannian surfaces B, existence has been proved in [10]. In particular, if the surface B happens to be simply connected then this result was already proven in [25].…”

Section: Critical Vertical Torimentioning

confidence: 93%

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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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Section: Correspondence Resultsmentioning

confidence: 94%

Section: Critical Vertical Torimentioning

confidence: 93%

Critical tori for mean curvature energies in Killing submersions

Pámpano

2020

Nonlinear Analysis

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“…Proposition 2.1. [2] Let M be a Riemannian surface and choose any r ∈ C ∞ (M). Then, there exists a Killing submersion over M with bundle curvature r. In particular, it can be chosen to have compact fibers.…”

Section: Killing Submersionsmentioning

confidence: 99%

“…From Theorem 1.1, by a case by case inspection, we easily deduce that a CMC proper biharmonic surface S in a Killing submersion E(G, r) with constant angle φ must be an Hopf tube, that is φ = π/2. 2…”

Section: Introductionmentioning

confidence: 99%

Biharmonic constant mean curvature surfaces in Killing submersions

Montaldo

Onnis²,

Passamani³

2018

Journal of Geometry and Physics

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A 3-dimensional Riemannian manifold is called Killing submersion if it admits a Riemannian submersion over a surface such that its fibers are the trajectories of a complete unit Killing vector field. In this paper, we give a characterization of proper biharmonic CMC surfaces in a Killing submersion. In the last part, we also classify the proper biharmonic Hopf cylinders in a Killing submersion.2000 Mathematics Subject Classification. 53C42, 58E20.1 is the curvature operator on (N, h). Equation (4) is a fourth order semi-linear elliptic system of differential equations. We also note that any harmonic map is an absolute minimum of the bienergy and, so, it is trivially biharmonic. Therefore, a general working plan is to study the existence of proper biharmonic maps, i.e. biharmonic maps which are not harmonic. We refer to [4,12] for existence results and general properties of biharmonic maps.In this paper, we study biharmonic surfaces in the total space of a Killing submersion. A Riemannian submersion π : E → M of a 3-dimensional Riemannian manifold E over a surface M will be called a Killing submersion if its fibers are the trajectories of a complete unit Killing vector field ξ. Killing submersions are determined by two functions on M, its Gaussian curvature G and the so called bundle curvature r (for more details, see [9,11] and Section 2) and, for this reason, we denote them by E(G, r). A remarkable family of Killing submersions are the homogeneous 3-spaces. In fact, with the exception of the hyperbolic 3-space H 3 (−1), simply-connected homogeneous Riemannian 3-manifolds with isometry group of dimension 4 or 6 can be represented by a 2-parameter family E(c, b), where c, b ∈ R. These E(c, b)-spaces are 3-manifolds admitting a global unit Killing vector field whose integral curves are the fibers of a certain Riemannian submersion over the simply-connected constant Gaussian curvature surface M(c) and, therefore, they determine Killing submersions π : E(c, b) → M(c) (for more details, see [6]). A local description of these examples can be given by using the so called Bianchi-Cartan-Vranceanu spaces (see Section 2). As it turns out, the E(c, b)-spaces are the only simply-connected homogeneous 3-manifolds admitting the structure of a Killing submersion (see [11]).

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“…Motivated by this fundamental fact, Killing submersions with 2-dimensional base and 1-dimensional fiber are now actively studied in differential geometry (see e.g. [9], [18], [33], [48]). It should be emphasize that Killing submersions with 2-dimensional base and 1-dimensional fiber involve geometrically natural dynamical systems -magnetic trajectories.…”

Section: Introductionmentioning

confidence: 99%