Description of the set of singular entire solutions of the maximal surface equation

Klyachin, Aleksey

doi:10.1070/sm2003v194n07abeh000753

Cited by 6 publications

(7 citation statements)

References 9 publications

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“…Similar result in the Lorentzian setting can be founded in [11]. Also a different approach to an end by Klyachin can be founded in [13]. We omit the proof.…”

Section: Dg Dzmentioning

confidence: 61%

Maximal Annuli With Parallel Planar Boundaries in the Three-Dimensional Lorentz–minkowski Space

Pyo

2010

Bull. Aust. Math. Soc.

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We prove that maximal annuli in L 3 bounded by circles, straight lines or cone points in a pair of parallel spacelike planes are part of either a Lorentzian catenoid or a Lorentzian Riemann's example. We show that under the same boundary condition, the same conclusion holds even when the maximal annuli have a planar end. Moreover, we extend Shiffman's convexity result to maximal annuli; but by using Perron's method we construct a maximal annulus with a planar end where a Shiffman-type result fails.2000 Mathematics subject classification: primary 53C42; secondary 53C50, 53A10.

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“…Similar result in the Lorentzian setting can be founded in [11]. Also a different approach to an end by Klyachin can be founded in [13]. We omit the proof.…”

Section: Dg Dzmentioning

confidence: 61%

Maximal Annuli With Parallel Planar Boundaries in the Three-Dimensional Lorentz–minkowski Space

Pyo

2010

Bull. Aust. Math. Soc.

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“…The proof follows from some well known classic results by Osserman [18], Jorge and Meeks [9] and R. Schoen [22] for minimal surfaces. A different approach in the Lorentzian setting can be found in [10]. We omit the proof.…”

Section: Global Behavior Of Complete Embedded Maximal Surfaces With I...mentioning

confidence: 99%

“…Theorem 2.2 (Klyachin [10]) Let h(x) be a function defined on a compact set K ⊂ {x 3 = 0}, Int(K) = ∅, and satisfying the inequality |h(x) − h(y)| < |x − y| for all x, y ∈ K, x = y. Then, for every timelike vector v there exists a unique solution u ∈ C 2 (R 2 − K) ∩ C(R 2 ) to the maximal graph equation such that u| K = h and F ∞ = v, where as above F ∞ is the flux at the end.…”

Section: Existence and Uniqueness Of Cmf Graphs With Any Number Of Si...mentioning

confidence: 99%

“…Klyachin and Miklyukov [11] have given results on the existence of solutions, with a finite number of isolated singularities, to the maximal hypersurface equation in L n+1 with prescribed boundary conditions. More recently Klyachin [10] has studied the existence, uniqueness and asymptotic behavior of entire maximal graphs in L n+1 with prescribed flux vector at infinity and compact singular set. A spacelike surface in L 3 is said to be an entire graph if its orthogonal projection over any spacelike plane is a homeomorphism.…”

Section: Introductionmentioning

confidence: 99%

“…More recently Klyachin [10] has studied the existence, uniqueness and asymptotic behavior of entire maximal graphs in L n+1 with prescribed flux vector at infinity and compact singular set. A spacelike surface in L 3 is said to be an entire graph if its orthogonal projection over any spacelike plane is a homeomorphism.…”

Section: Introductionmentioning

confidence: 99%

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The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space

2005

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We prove that a complete embedded maximal surface in L 3 with a finite number of singularities is an entire maximal graph with conelike singularities over any spacelike plane, and so, it is asymptotic to a spacelike plane or a half catenoid.We show that the moduli space Gn of entire maximal graphs over {x3 = 0} in L 3 with n + 1 ≥ 2 singular points and vertical limit normal vector at infinity is a 3n + 4-dimensional differentiable manifold. The convergence in Gn means the one of conformal structures and Weierstrass data, and it is equivalent to the uniform convergence of graphs on compact subsets of {x3 = 0}. Moreover, the position of the singular points in R 3 and the logarithmic growth at infinity can be used as global analytical coordinates with the same underlying topology. We also introduce the moduli space Mn of marked graphs with n + 1 singular points (a mark in a graph is an ordering of its singularities), which is a (n + 1)-sheeted covering of Gn. We prove that identifying marked graphs differing by translations, rotations about a vertical axis, homotheties or symmetries about a horizontal plane, the corresponding quotient spacê Mn is an analytic manifold of dimension 3n−1. This manifold can be identified with a spinorial bundle Sn associated to the moduli space of Weierstrass data of graphs in Gn. Notations and Preliminary results .Since M is spacelike, then |g| = 1 on M.Remark 2.1 For convenience, we also deal with surfaces M having ∂(M ) = ∅, and in this case, we always suppose that φ 3 and g extend analitically beyond ∂M.Conversely, let M, g and φ 3 be a Riemann surface with possibly non empty boundary, a meromorphic map on M and an holomorphic 1-form φ 3 on M, such that |g(P )| = 1, ∀P ∈ M, and the 1-forms φ j , j = 1, 2, 3 defined as above are holomorphic, have no real periods and have no common zeroes.

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A Sharp Gradient Estimate and $$W^{2,q}$$ Regularity for the Prescribed Mean Curvature Equation in the Lorentz-Minkowski Space

Bonheure,

Iacopetti

2023

Arch Rational Mech Anal

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We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namelywhere N 3. We first prove a new gradient estimate for classical solutions with smooth data ρ. As a consequence, we obtain that the unique weak solution of the equation satisfying a homogeneous boundary condition at infinity is locally of class W 2,q and strictly spacelike in R N , provided that ρ ∈ L q (R N ) ∩ L m (R N ) with q > N and m ∈ [1, 2NN +2 ]. Contents

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Description of the set of singular entire solutions of the maximal surface equation

Cited by 6 publications

References 9 publications

Maximal Annuli With Parallel Planar Boundaries in the Three-Dimensional Lorentz–minkowski Space

Maximal Annuli With Parallel Planar Boundaries in the Three-Dimensional Lorentz–minkowski Space

The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space

A Sharp Gradient Estimate and $$W^{2,q}$$ Regularity for the Prescribed Mean Curvature Equation in the Lorentz-Minkowski Space

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