Dense solutions to the Cauchy problem for minimal surfaces

Gálvez, José A.; Mira, Pablo

doi:10.1007/s00574-004-0021-z

Cited by 15 publications

(10 citation statements)

References 5 publications

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“…The solution is then given by analytic extension. The geometric Cauchy problem has recently been studied in several situations involving holomorphic representations of surface classes -see, for example, [14,15,2,10,13,21,4,9,17,6]. The solution of this problem is clearly a useful tool, both for proving general local properties of the surfaces and for constructing interesting examples.…”

Section: Introductionmentioning

confidence: 99%

The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps

Brander¹,

Svensson²

2013

J. Differential Geom.

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The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-space, and for timelike constant non-zero mean curvature (CMC) surfaces in the Lorentz-Minkowski 3space. We prove that there is a unique solution if the prescribed curve is non-characteristic, and for characteristic initial curves (asymptotic curves for pseudospherical surfaces and null curves for timelike CMC) it is necessary and sufficient for similar data to be prescribed along an additional characteristic curve that intersects the first. The proofs also give a means of constructing all solutions using loop group techniques. The method used is the infinite dimensional d'Alembert type representation for surfaces associated with Lorentzian harmonic maps (1-1 wave maps) into symmetric spaces, developed since the 1990's. Explicit formulae for the potentials in terms of the prescribed data are given, and some applications are considered.2000 Mathematics Subject Classification. Primary 53A05, 53A35; Secondary 53A10, 53C42, 53C43.

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Section: Introductionmentioning

confidence: 99%

The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps

Brander¹,

Svensson²

2013

J. Differential Geom.

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“…The first example of such, due to Rosenberg, was obtained by Schwarzian reflection on a fundamental domain, is simply-conneted, and has bounded curvature. Later, Gálvez and Mira [24] found complete dense simply-connected minimal surfaces in R 3 , in explicit coordinates, as solution to certain Björling problems. Finally, López [30] constructed complete dense minimal surfaces in R 3 with weak finite total curvature, arbitrary genus, and parabolic conformal type; so far, these are the only known examples with non-trivial topology.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Complete minimal surfaces densely lying in arbitrary domains of ℝn

Alarcón

Castro-Infantes

2017

Geom. Topol.

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In this paper we prove that, given an open Riemann surface M and an integer n ≥ 3, the set of complete conformal minimal immersions M → R n with X(M ) = R n forms a dense subset in the space of all conformal minimal immersions M → R n endowed with the compact-open topology. Moreover, we show that every domain in R n contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface.Our method of proof can be adapted to give analogous results for non-orientable minimal surfaces in R n (n ≥ 3), complex curves in C n (n ≥ 2), holomorphic null curves in C n (n ≥ 3), and holomorphic Legendrian curves in C 2n+1 (n ∈ N).

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“…The solution to this problem was obtained by Schwarz in 1890, by using holomorphic data. Some extensions of this classical Björling problem to other geometric theories, as well as some global applications of it have been developed in [1,2,[5][6][7]. In particular, the extension of the Björling problem to R n consists on the following:…”

Section: Introductionmentioning

confidence: 99%

Complete minimal Möbius strips in and the Björling problem

Mira

2006

Journal of Geometry and Physics

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Dense solutions to the Cauchy problem for minimal surfaces

Cited by 15 publications

References 5 publications

The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps

The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps

Complete minimal surfaces densely lying in arbitrary domains of ℝn

Complete minimal Möbius strips in and the Björling problem

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