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Albujer, Alma L.; Caballero, Magdalena

doi:10.1016/j.jmaa.2016.07.062

Cited by 17 publications

(15 citation statements)

References 12 publications

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“…However, there is no known classification of such hypersurfaces similar to Kobayashi's result. In [1] the authors have shown that those hypersurfaces do not have elliptic points and have obtained several interesting consequences about the geometry of such hypersurfaces, generalizing some results in [2].…”

Section: Introductionmentioning

confidence: 71%

“…Finally, we generalize some results on the graphs of the solutions which are not a consequence of the non-existence of elliptic points, specifically Lemma 7, Theorem 8 and Corollary 1 from [2].…”

Section: Introductionmentioning

confidence: 79%

“…He showed that the graphs of the solutions to (1.1) with H R = H L = 0 are open pieces of a spacelike plane or of a helicoid, in the region where the helicoid is a spacelike surface. Recently, Albujer, Caballero and Sánchez [2,3] have continued with the study of spacelike surfaces with the same mean curvature in R 3 and in L 3 , not necessarily constant. On one hand, they have shown that the Gaussian curvature in R 3 of those surfaces is always non-positive and have obtained several interesting consequences about the geometry of such surfaces.…”

Section: Introductionmentioning

confidence: 99%

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On the Solutions to the $H_R H_L$ Hypersurface Equation

Albujer¹,

Caballero²,

Alarcón³

2017

Proceedings Book of International Workshop on Theory of Submanifolds

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Abstract.Spacelike hypersurfaces in the Lorentz-Minkowski (n+1)-dimensional space L n+1 can be endowed with another Riemannian metric, the one induced by the Euclidean space R n+1 . The hypersurfaces with the same mean curvature with respect to both metrics can be locally determined by a smooth function u satisfying |Du| < 1, and being the solution to a certain partial differential equation. We call this equation the H R = H L hypersurface equation. In the particular case in which n = 2 and both curvatures vanish, Kobayashi proved that the graphs determined by the solutions of such equation are open pieces of spacelike planes or helicoids, in the region where they are spacelike. In this manuscript we prove the existence of a family of solutions whose graphs have non-zero mean curvature, and we present an inequality relating the mean curvature to the width of the domain of certain solutions, those without critical points.

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Section: Introductionmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

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On the Solutions to the $H_R H_L$ Hypersurface Equation

Albujer¹,

Caballero²,

Alarcón³

2017

Proceedings Book of International Workshop on Theory of Submanifolds

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“…Since the curvature of a curve and the mean curvature of an n-dimensional hypersurface are important invariants for curves and surfaces, many authors have studied these notions for different types of curves and surfaces for a long time in different spaces, such as Euclidean, Minkowski, Galilean and pseudo-Galilean spaces ( [2], [4], [9], [11], [14], [15], [19], [21] and etc. ).…”

Section: Introductionmentioning

confidence: 99%

Ruled and Rotational Surfaces Generated by Non-Null Curves with Zero Weighted Curvature in $(\mathbb{L}^{3},ax^{2}+by^{2})$

Altin

Kazan

Karadağ

2020

International Electronic Journal of Geometry

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In this study, firstly we give the weighted curvatures of non-null planar curves in Lorentz-Minkowski space with density e ax 2 +by 2 and obtain the planar curves whose weighted curvatures vanish in this space under the condition that the constants a and b are not zero at the same time. After giving the Frenet vectors of the non-null planar curves with zero weighted curvature in Lorentz-Minkowski space with density e ax 2 , we create the Smarandache curves of them. With the aid of these curves and their Smarandache curves, we get the ruled surfaces whose base curves are non-null curves of which vanishing weighted curvature and ruling curves are Smarandache curves of them. Followingly, we give some characterizations for these ruled surfaces by obtaining the mean and Gaussian curvatures, distribution parameters and striction curves of them. Also, rotational surfaces which are generated by non-null planar curves with zero weighted curvatures in Lorentz-Minkowski space E 3 1 with density e ax 2 +by 2 are studied under the condition that the constants a and b are not zero at the same time. We draw the graphics of the obtained surfaces.

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“…In [16], Kobayashi pointed out that the only spacelike surface satisfying H R = H L = 0 for dimension two is an open piece of a spacelike plane or of a spacelike helicoid in the region. 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].…”

Section: Introductionmentioning

confidence: 99%