Height and Area Estimates for Constant Mean Curvature Graphs in $$\mathbb {E}(\kappa ,\tau )$$ E ( κ , τ ) -Spaces

Manzano, José M.; Nelli, Barbara

doi:10.1007/s12220-017-9810-7

Cited by 13 publications

(16 citation statements)

References 33 publications

(85 reference statements)

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“…In the case of

E (κ, τ)

spaces with

κ \leq 0

we can use the model for

E (κ, τ)

as (see [11]) the space

E false(κ, τ false) = \{(x, y, z) \in R^{3} : 1 + \frac{κ}{4} true(x^{2} + y^{2} true) > 0\}

endowed with the Riemannian metric such that the following three vector fields

\begin{matrix} true \\ left E_{1} = ([] 1 + \frac{κ}{4} (x^{2} + y^{2})) \frac{\partial}{\partial x} - τ y \frac{\partial}{\partial z}, \\ left E_{2} = ([] 1 + \frac{κ}{4} (x^{2} + y^{2})) \frac{\partial}{\partial y} + τ x \frac{\partial}{\partial z}, \\ left E_{3} = \frac{\partial}{\partial z} \end{matrix}

constitutes an orthonormal basis in each tangent space. Observe that

π (x, y, z) \to (x, y)

is a Riemannian submersion from

E (κ, τ)

{double-struckM}^{2} false(κ false)

whose fibers are the integral curves of the unit‐length Killing vector field E 3 .…”

Section: Resultsmentioning

confidence: 99%

Mean curvature of hypersurfaces in Killing submersions with bounded shadow

Gimeno

2020

Mathematische Nachrichten

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Given a complete hypersurface isometrically immersed in an ambient manifold, in this paper we provide a lower bound for the norm of the mean curvature vector field of the immersion assuming that: 1)The ambient manifold admits a Killing submersion with unit‐length Killing vector field. 2)The projection of the image of the immersion is bounded in the base manifold. 3)The hypersurface is stochastically complete, or the immersion is proper.

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“…In the case of

E (κ, τ)

spaces with

κ \leq 0

we can use the model for

E (κ, τ)

as (see [11]) the space

E false(κ, τ false) = \{(x, y, z) \in R^{3} : 1 + \frac{κ}{4} true(x^{2} + y^{2} true) > 0\}

endowed with the Riemannian metric such that the following three vector fields

\begin{matrix} true \\ left E_{1} = ([] 1 + \frac{κ}{4} (x^{2} + y^{2})) \frac{\partial}{\partial x} - τ y \frac{\partial}{\partial z}, \\ left E_{2} = ([] 1 + \frac{κ}{4} (x^{2} + y^{2})) \frac{\partial}{\partial y} + τ x \frac{\partial}{\partial z}, \\ left E_{3} = \frac{\partial}{\partial z} \end{matrix}

constitutes an orthonormal basis in each tangent space. Observe that

π (x, y, z) \to (x, y)

is a Riemannian submersion from

E (κ, τ)

{double-struckM}^{2} false(κ false)

whose fibers are the integral curves of the unit‐length Killing vector field E 3 .…”

Section: Resultsmentioning

confidence: 99%

Mean curvature of hypersurfaces in Killing submersions with bounded shadow

Gimeno

2020

Mathematische Nachrichten

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“…No monotonicity formula for minimal surfaces in Nil 3 (τ ) has been found so far, which is one of the key techniques leading to the same result in R 3 , and Theorem 4.4 is the first lower bound for the area growth for entire minimal graphs in Nil 3 (τ ). A sharp upper bound was found by the author and Nelli [28], showing that such area growth is at most cubic.…”

Section: Introductionmentioning

confidence: 82%

“…Remark 4.5. Equation (4.3) does not give much information about the integrability of K, since ν 2 is never an integrable function on an entire minimal graph [28], and ν 4 can be either integrable or not, as we shall see in the next examples.…”

Section: )mentioning

confidence: 99%

Dual quadratic differentials and entire minimal graphs in Heisenberg space

Manzano

2018

Ann Glob Anal Geom

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We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces L(κ, τ ) with isometry group of dimension 4, which are dual to the Abresch-Rosenberg differentials in the Riemannian counterparts E(κ, τ ), and obtain some consequences. On the one hand, we give a very short proof of the Bernstein problem in Heisenberg space, and provide a geometric description of the family of entire graphs sharing the same differential in terms of a 2-parameter conformal deformation. On the other hand, we prove that entire minimal graphs in Heisenberg space have negative Gauss curvature.

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“…Notice that a vertical Euclidean plane is also a minimal surface and it is flat. This means that half of the catenoid is a graph over the exterior domain r r 0 with zero boundary values [14]. One finds a detailed study of the vertical catenoids in [2].…”

Section: Examples Of Complete Minimal Graphsmentioning

confidence: 99%

“…Two questions arises about solutions on a strip. Note that, by [14,Theorem 7], any non trivial solution of the minimal surface equation, with zero boundary value on a strip has at least linear growth (see Definition 4.2). In the same article Manzano and Nelli prove that the growth of an entire minimal graph in N il 3 has order at most three (Theorem 6).…”

mentioning

confidence: 99%

Minimal graphs in $$Nil_3:$$ N i l 3 : existence and non-existence results

2017

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We study the minimal surface equation in the Heisenberg space, N il3. A geometric proof of non existence of minimal graphs over non convex, bounded and unbounded domains is achieved (our proof holds in the Euclidean space as well). We solve the Dirichlet problem for the minimal surface equation over bounded and unbounded convex domains, taking bounded, piecewise continuous boundary value. We are able to construct a Scherk type minimal surface and we use it as a barrier to construct non trivial minimal graphs over a wedge of angle θ ∈ [ π 2 , π[ taking non negative continuous boundary data, having at least quadratic growth. In the case of an halfplane, we are also able to give solutions (with either linear or quadratic growth), provided some geometric hypothesis on the boundary data. Finally, some open problem arising from our work, are posed.

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Height and Area Estimates for Constant Mean Curvature Graphs in $$\mathbb {E}(\kappa ,\tau )$$ E ( κ , τ ) -Spaces

Cited by 13 publications

References 33 publications

Mean curvature of hypersurfaces in Killing submersions with bounded shadow

Mean curvature of hypersurfaces in Killing submersions with bounded shadow

Dual quadratic differentials and entire minimal graphs in Heisenberg space

Minimal graphs in $$Nil_3:$$ N i l 3 : existence and non-existence results

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