The structure of minimal surfaces in CAT(0) spaces

Stadler, Stephan

doi:10.48550/arxiv.1808.06410

Cited by 6 publications

(6 citation statements)

References 26 publications

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“…However the constant β will depend on κ(Γ) in this case. To achieve this one performs the same proof using the funnel extension discussed in section 3.1 of [Sta18] instead of X Γ . 4.2.…”

Section: Theorem 34 ([Lw16]mentioning

confidence: 99%

Plateau's problem for singular curves

Creutz¹

2019

Preprint

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Section: Theorem 34 ([Lw16]mentioning

confidence: 99%

Plateau's problem for singular curves

Creutz¹

2019

Preprint

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“…On any sector, we fix an orientation so that the left leg and the right leg of S α,r are defined. The following lemma generalizes [Sta,Lemma 21] to spaces satisfying positive upper curvature bounds.…”

Section: Interior Lipschitz Regularitymentioning

confidence: 87%

The Plateau-Douglas problem for singular configurations and in general metric spaces

Creutz¹,

Fitzi²

2020

Preprint

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Assume you are given a finite configuration Γ of disjoint rectifiable Jordan curves in R n . The Plateau-Douglas problem asks whether there exists a minimizer of area among all compact surfaces of genus at most p which span Γ. While the solution to this problem is well-known, the classical approaches break down if one allows for singular configurations Γ where the curves are potentially non-disjoint or self-intersecting. Our main result solves the Plateau-Douglas problem for such potentially singular configurations. Moreover, our proof works not only in R n but in general proper metric spaces. Thus we are also able to extend previously known existence results of Jürgen Jost as well as of the second author together with Stefan Wenger for regular configurations. In particular, existence is new for disjoint configurations of Jordan curves in general complete Riemannian manifolds. A minimal surface of fixed genus p bounding a given configuration Γ need not always exist, even in the most regular settings. Concerning this problem, we also generalize the approach for singular configurations via minimal sequences satisfying conditions of cohesion and adhesion to the setting of metric spaces.

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“…In [14], Alexander Lytchak and the second author used them to deform general CAT(0) spaces (minimal discs). In [25], the second author used them in the proof a CAT(0) version of the Fary-Milnor theorem to control the mapping behavior of minimal surfaces (minimal discs).…”

Section: Introductionmentioning

confidence: 99%