Plateau's problem for singular curves

Creutz, Paul

doi:10.48550/arxiv.1904.12567

Cited by 3 publications

(8 citation statements)

References 11 publications

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“…The solution for singular configurations is new even in R n . Theorem 1.2 also generalizes the main results of [18] and [10] as we are able to drop the assumption that X admits a local quadratic isoperimetric inequality. In particular, the existence is new for regular configurations in complete Riemannian manifolds which might not be homogeneously regular.…”

Section: Introductionsupporting

confidence: 59%

“…Indeed the classical methods would necessarily produce harmonic area minimizers but simple examples like the figure-eight curve in R 2 show that not every self-intersecting curve can bound such disks. Still the generality of [28] and a simple extension trick allowed the first author to solve the Plateau problem for possibly self-intersecting curves in proper metric spaces which satisfy a local quadratic isoperimetric inequality [10]. In R n this improved a previous existence result due to Hass [20].…”

Section: Introductionmentioning

confidence: 96%

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The Plateau-Douglas Problem for Singular Configurations and in General Metric Spaces

Creutz

Fitzi

2023

Arch Rational Mech Anal

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Assume you are given a finite configuration $$\Gamma $$ Γ of disjoint rectifiable Jordan curves in $${\mathbb {R}}^n$$ R n . The Plateau-Douglas problem asks whether there exists a minimizer of area among all compact surfaces of genus of at most p which span $$\Gamma $$ Γ . While the solution to this problem is well-known, the classical approaches break down if one allows for singular configurations $$\Gamma $$ 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 $${\mathbb {R}}^n$$ R n but in general proper metric spaces. In particular, the existence of an area minimizer is new for disjoint configurations of Jordan curves in general complete Riemannian manifolds. A minimal surface of fixed genus p bounding a given configuration $$\Gamma $$ 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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Section: Introductionsupporting

confidence: 59%

Section: Introductionmentioning

confidence: 96%

The Plateau-Douglas Problem for Singular Configurations and in General Metric Spaces

Creutz

Fitzi

2023

Arch Rational Mech Anal

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“…The existence of solutions to the Plateau problem spanning self-intersecting boundaries has been addressed in [21], whose results have been recently improved in [10]. Without entering deeply into the details, it is known that, depending on the geometry of γ (in this case, depending on the distance between the two circles C 1 and C 2 ) two kind of solutions are expected:…”

Section: A Plateau Problem For a Self-intersecting Boundary Space Curvementioning

confidence: 99%

“…X(L 2 ) = P 2 are the two endpoints of γ 0 , and X restricted to the sectors between L i , i = 1, 2, and ∂B 1 parametrizes the disc filling C i , i = 1, 2. Moreover the map X can be still taken Sobolev regular (see [10] for details).…”

Section: A Plateau Problem For a Self-intersecting Boundary Space Curvementioning

confidence: 99%

The $L^1$-relaxed area of the graph of the vortex map

Bellettini,

Elshorbagy,

Scala

2021

Preprint

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“…The tricks in both proofs show that it is useful to consider singular spaces even in the case when the original space U is smooth; this is a powerful freedom of Alexandrov's world. More involved examples of such arguments are given by Dmitry Burago, Sergei Ferleger, and Alexey Kanonenko [13], Paul Creutz [14], and Stephan Stadler [15].…”

Section: Introductionmentioning

confidence: 99%

Conformal deformations of CAT(0) spaces

Lytchak

Stadler

2018

Math. Ann.

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We construct short retractions of a CAT(1) space to its small convex subsets. This construction provides an alternative geometric description of an analytic tool introduced by Wilfrid Kendall.Our construction uses a tractrix flow which can be defined as a gradient flow for a family of functions of certain type. In an appendix we prove a general existence result for gradient flows of time-dependent locally Lipschitz semiconcave functions, which is of independent interest.Proof. Note that geodesics [α(t)p] lie in Dom f t for any t.Since f t is λ-concave, we haveHence the only-if part follows. Given a point p ∈ U and t, consider a pointp ∈ [α(t)p]. Applying ➎ forp and the triangle inequality, we get

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Plateau's problem for singular curves

Cited by 3 publications

References 11 publications

The Plateau-Douglas Problem for Singular Configurations and in General Metric Spaces

The Plateau-Douglas Problem for Singular Configurations and in General Metric Spaces

The $L^1$-relaxed area of the graph of the vortex map

Conformal deformations of CAT(0) spaces

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