Quasi-Minimal Lagrangian Surfaces Whose Mean Curvature Vectors Are Eigenvectors

Sasahara, Toru

doi:10.1515/dema-2005-0121

Cited by 18 publications

(12 citation statements)

References 16 publications

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“…By using a − b = b + c = −c − d = α we can prove the lemma (cf. Lemma 8 of [16] REMARK 10. In [6], marginally trapped biharmonic surfaces of real index 1 of C 2 1 were classified.…”

Section: Biharmonic Lagrangian Surfacesmentioning

confidence: 98%

“…In case of (II), by (4.1), (4.31) and (4.33) we see that a Legendrian liftf ∈ C 3 2 satisfies the following PDE system:f xx = bif y +f , (4.42) Case (III): H, H = 0 and H = 0. In this case, the author [16] proved that M is biharmonic if and only if = 0 and H = 0, however this fact is not enough information for classification of such surfaces. So, we need to investigate necessary and sufficient conditions for M to be biharmonic more precisely.…”

Section: Biharmonic Lagrangian Surfacesmentioning

confidence: 99%

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Biharmonic Lagrangian Surfaces of Constant Mean Curvature in Complex Space Forms

Sasahara¹

2007

Glasgow Math. J.

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A natural generalization of the class of minimal submanifolds from the variational point of view is the one of biharmonic submanifolds (see, [9]), which was introduced by Eells and Sampson [9]. Thus, it is worthwhile and interesting to investigate biharmonic Lagrangian submanifolds. For recent developments in the study of biharmonic submanifolds see, for example, [2], [3], [10] and [15]. In this paper, biharmonic Lagrangian surfaces of constant mean curvature in complex space forms are classified. In particular, we obtain new examples of marginally trapped Lagrangian surfaces in an indefinite complex Euclidean plane. This implies that the classification of marginally trapped biharmonic surfaces due to Chen and Ishikawa [6] is not complete.

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“…By using a − b = b + c = −c − d = α we can prove the lemma (cf. Lemma 8 of [16] REMARK 10. In [6], marginally trapped biharmonic surfaces of real index 1 of C 2 1 were classified.…”

Section: Biharmonic Lagrangian Surfacesmentioning

confidence: 98%

Section: Biharmonic Lagrangian Surfacesmentioning

confidence: 99%

Biharmonic Lagrangian Surfaces of Constant Mean Curvature in Complex Space Forms

Sasahara¹

2007

Glasgow Math. J.

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“…Quasi-minimal Lagrangian surfaces in a Lorentzian complex space form whose mean curvature vector H satisfies ∆H = λH for some constant λ were studied by Sasahara in [34].…”

Section: Quasi-minimal Lagrangian Surfacesmentioning

confidence: 99%

Black holes, marginally trapped surfaces and quasi-minimal surfaces

Chen¹

2009

Tamkang J. Math.

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The concept of trapped surfaces introduced by Sir Roger Penrose in [Phys. Rev. Lett. 14 (1965), 57-59] plays an extremely important role in cosmology and general relativity. A black hole is a trapped region in a space-time enclosed by a marginally trapped surface. In term of mean curvature vector, a space-like surface in a space-time is marginally trapped if its mean curvature vector field is light-like at each point. In this article, we survey recent classification results on marginally trapped surfaces from differential geometric viewpoint. Also, we survey recent results on a closely related subject; namely, quasi-minimal surfaces in pseudo-Riemannian manifolds.

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“…−, e 1 ) = βF e 2 , h(e 1 , e 2 ) = F e 1 , h(e 2 , e 2 ) = αF e 1 + F e 2 (5 16). for some real-valued functions α, β, From Lemma 5.1 we haveω 1 = 0, e 1 α = ω 2 − 2 sinh θ, e 2 β = 3βω 2 − 2β sinh θ. that the Gauss curvature K of M satises K = αβ cosh 2 θ.In this case, (5.18) gives K = 0.…”

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confidence: 93%

“…(Quasi-minimal surfaces are also known as marginally trapped surfaces in general relativity.) Among others, quasi-minimal Lorentzian surfaces in Lorentzian complex space forms have been studied in [7,8,9,16,17].…”

Section: Introductionmentioning

confidence: 99%

Classification of quasi-minimal slant surfaces in Lorentzian complex space forms

Chen

Iordache

2008

Acta Math Hung

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From J-action point of views, slant surfaces are the simplest and the most natural surfaces of a (Lorentzian) Kähler surface (M,g, J). Slant surfaces arise naturally and play some important roles in the studies of surfaces of Kähler surfaces (see, for instance, [13]). In this article, we classify quasi-minimal slant surfaces in the Lorentzian complex plane C 2 1 . More precisely, we prove that there exist ve large families of quasi-minimal proper slant surfaces in C 2 1 . Conversely, quasi-minimal slant surfaces in C 2 1 are either Lagrangian or locally obtained from one of the ve families. Moreover, we prove that quasi-minimal slant surfaces in a non-at Lorentzian complex space form are Lagrangian.

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Quasi-Minimal Lagrangian Surfaces Whose Mean Curvature Vectors Are Eigenvectors

Abstract: Abstract. We investigate quasi-minimal Lagrangian surfaces whose mean curvature vectors sire eigenvectors of the Laplace operator.

Cited by 18 publications

References 16 publications

Biharmonic Lagrangian Surfaces of Constant Mean Curvature in Complex Space Forms

Biharmonic Lagrangian Surfaces of Constant Mean Curvature in Complex Space Forms

Black holes, marginally trapped surfaces and quasi-minimal surfaces

Classification of quasi-minimal slant surfaces in Lorentzian complex space forms

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