Spacelike Hypersurfaces in the Lorentz-Minkowski Space with the Same Riemannian and Lorentzian Mean Curvature

Alarcón, Eva M.; Albujer, Alma L.; Caballero, Magdalena

doi:10.1007/978-3-319-66290-9_1

Cited by 7 publications

(4 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, Albujer and Caballero [3] studied the H R = H L surface equation and in particular they proved that the only spacelike graphs in L 3 , defined over a domain Ω ⊆ R 2 of infinite width, satisfying H R = H L and asymptotic to a spacelike plane are (pieces of ) spacelike planes. This result holds true for higher dimension, for more details, see [2]. Affine functions (whose gradient has norm less than 1) are clearly solutions to (1.3).…”

Section: Introductionmentioning

confidence: 62%

Multiple radial solutions for Dirichlet problem involving two mean curvature equations in Euclidean and Minkowski spaces

Wang

2019

Bound Value Probl

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 62%

Multiple radial solutions for Dirichlet problem involving two mean curvature equations in Euclidean and Minkowski spaces

Wang

2019

Bound Value Probl

View full text Add to dashboard Cite

show abstract

“…called the H R = H L hypersurface equation. This equation is a quasilinear elliptic partial differential equation, everywhere except at those points at which Du vanishes, where the equation is parabolic, see [1].…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Albujer, Caballero and Sánchez [3,4] have continued with the study of spacelike surfaces with the same mean curvature in R 3 and in L 3 , not necessarily constant. The generalization of some of its results to general dimension has been given in [1,2].…”

Section: Introductionmentioning

confidence: 99%

On the hypersufaces of the Euclidean space which are simultaneously minimal and maximal

Caballero¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

It is well known that the only surfaces that are simultaneously minimal in R 3 and maximal in L 3 are open pieces of helicoids (in the region in which they are spacelike) and of spacelike planes, [11]. The proof of this result consists in showing that the level curves of those surfaces are lines, and so the surfaces are ruled. And it finishes comparing the classification of minimal ruled surfaces to that of maximal ruled surfaces.In this manuscript we consider the general case of spacelike hypersurfaces in the (n + 1)-dimensional Euclidean space which are simultaneously maximal and minimal. We show that its level curves are minimal hypersurfaces in the n-dimensional Euclidean space.

show abstract

“…We call them mean isocurved. These hypersurfaces have been studied by Albujer-Caballero [4] in the case where the ambient space is L 3 (see [1] as well). Actually, during the preparation of this paper, we became acquainted with the recent works by Alarcn-Alias-Santos [2] and Albujer-Caballero [3] which have some overlapping with ours in this subject.…”

Section: Introductionmentioning

confidence: 99%

Helicoids and Catenoids in $M\times\mathbb{R}$

Lima¹,

Roitman²

2019

Preprint

View full text Add to dashboard Cite

Given an arbitrary C ∞ Riemannian manifold M n , we consider the problem of introducing and constructing minimal hypersurfaces in M ×R which have the same fundamental properties of the standard helicoids and catenoids of Euclidean space R 3 = R 2 × R. Such hypersurfaces are defined by imposing conditions on their height functions and horizontal sections, and then called vertical helicoids and vertical catenoids. We establish that vertical helicoids in M ×R have the same fundamental uniqueness properties of the helicoids in R 3 . We provide several examples of vertical helicoids in the case where M is one of the simply connected space forms. Vertical helicoids which are entire graphs of functions on Nil 3 and Sol 3 are also presented. We give a local characterization of hypersurfaces of M × R which have the gradient of their height functions as a principal direction. As a consequence, we prove that vertical catenoids exist in M × R if and only if M admits families of isoparametric hypersurfaces. If so, they can be constructed through the solutions of a certain first order linear differential equation. Finally, we give a complete classification of the hypersurfaces of M × R whose angle function is constant.

show abstract

Spacelike Hypersurfaces in the Lorentz-Minkowski Space with the Same Riemannian and Lorentzian Mean Curvature

Cited by 7 publications

References 15 publications

Multiple radial solutions for Dirichlet problem involving two mean curvature equations in Euclidean and Minkowski spaces

Multiple radial solutions for Dirichlet problem involving two mean curvature equations in Euclidean and Minkowski spaces

On the hypersufaces of the Euclidean space which are simultaneously minimal and maximal

Helicoids and Catenoids in $M\times\mathbb{R}$

Contact Info

Product

Resources

About