Corpus ID: 202539545

Duality of boundary value problems for minimal and maximal surfaces

@article{Akamine2019DualityOB,
  title={Duality of boundary value problems for minimal and maximal surfaces},
  author={Shintaro Akamine and Hiroki Fujino},
  journal={arXiv: Differential Geometry},
  year={2019}
}
In 1966, Jenkins and Serrin gave existence and uniqueness results for infinite boundary value problems of minimal surfaces in the Euclidean space, and after that such solutions have been studied by using the univalent harmonic mapping theory. In this paper, we show that there exists a one-to-one correspondence between solutions of infinite boundary value problems for minimal surfaces and those of lightlike line boundary problems for maximal surfaces in the Lorentz-Minkowski spacetime. We also… Expand

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References

SHOWING 1-10 OF 36 REFERENCES
How many maximal surfaces do correspond to one minimal surface?
Abstract We discuss the minimal-to-maximal correspondence between surfaces and show that, under this correspondence, a congruence class of minimal surfaces in 3 determines an 2-family of congruenceExpand
Unstable Minimal Surfaces
Here it is shown that the existence of two minimal surfaces in a closed rectifiable contour Γ which are local minimizers of Dirichlet’s integral D guarantees the existence of a third minimal surfaceExpand
Prescribing singularities of maximal surfaces via a singular Björling representation formula
Abstract We derive a proper formulation of the singular Bjorling problem for spacelike maximal surfaces with singularities in the Lorentz–Minkowski 3-space which roughly asks whether there exists aExpand
The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions
We prove existence of Schoen's and other triply periodic minimal surfaces via conjugate (polygonal) Plateau problems. The simpler of these minimal surfaces can be deformed into constant meanExpand
Variational problems of minimal surface type II. Boundary value problems for the minimal surface equation
in a convex domain D, and taking on assigned continuous values on the boundary of D. This problem was solved by RAD6 in 1930, on the basis of the existence theorem for the parametric problem of leastExpand
Generalized maximal surfaces in Lorentz-Minkowski space L3
In this paper we carry out a systematic study of generalized maximal surfaces in Lorentz–Minkowski space L 3 , with emphasis on their branch points. Roughly speaking, such a surface is given by aExpand
Maximal surfaces of Riemann type in Lorentz-Minkowski space L3.
We classify the family of spacelike maximal surfaces in Lorentz-Minkowski 3-space L 3 which are foliated by pieces of circles. This space contains a curve of singly periodic maximal surfaces R thatExpand
Generalized Calabi correspondence and complete spacelike surfaces
We construct a twin correspondence between graphs with prescribed mean curvature in three-dimensional Riemannian Killing submersions and spacelike graphs with prescribed mean curvature inExpand
On univalent harmonic mappings and minimal surfaces
If S is the graph of a minimal surface, then when given parametrically by the Weierstrass representation, the first two coordinate functions give a univalent harmonic mapping. In this paper, theExpand
On the boundary behavior of orientation-preserving harmonic mappings
Nonconstant soluliuns of the partial differential equation where a is analytic in the open unit disk and |a| <1, are orientation-preserving harmonic mappings. We consider the case where ais a finiteExpand
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