The Plateau problem for polygonal boundary curves in Minkowski 3-space

Laura Desideri

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The Plateau problem for polygonal boundary curves in Minkowski 3-space

Laura Desideri

Differential Geometry · Submitted 16 Dec 2010

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that any given spacelike polygonal curve in generic position in Minkowski 3-space bounds at least one maxface of disk-type.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,507 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.68 · states limitation 0.72 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.95 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We apply Garnier's method to solve the Plateau problem for maximal surfaces in Minkowski 3-space. Our study relies on the improved version we gave of R. Garnier's resolution of the Plateau problem for polygonal boundary curves in Euclidean 3-space. Since in Minkowski space the method does not allow us to avoid the existence of singularities, the appropriate framework is to consider maxfaces -- generalized maximal surfaces without branch points, introduced by M. Umehara and K. Yamada. We prove that any given spacelike polygonal curve in generic position in Minkowski 3-space bounds at least one maxface of disk-type. This is a new result, since the only known result for the Plateau problem in Minkowski space (due to N. Quien) deals with boundary curves of regularity $C^{3,\alpha}$.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We apply Garnier's method to solve the Plateau problem for maximal surfaces in Minkowski 3-space.	contribution 0.61	0.90	0.10	asserted 1.00
s1	Our study relies on the improved version we gave of R.	method 0.64	0.83	0.03	asserted 0.92
s2	Garnier's resolution of the Plateau problem for polygonal boundary curves in Euclidean 3-space.	background 0.71	0.38	0.03	asserted 0.65
s3	Since in Minkowski space the method does not allow us to avoid the existence of singularities, the appropriate framework is to consider maxfaces -- generalized maximal surfaces without branch points, introduced by M. Umehara and K. Yamada.	method 0.52	0.24	0.05	asserted 1.00
s4	We prove that any given spacelike polygonal curve in generic position in Minkowski 3-space bounds at least one maxface of disk-type.	result 0.98	0.95	0.90	asserted 1.00
s5	This is a new result, since the only known result for the Plateau problem in Minkowski space (due to N.	gap 0.87	0.32	0.09	asserted 0.98
s6	Quien) deals with boundary curves of regularity $C^{3,\alpha}$.	gap 0.75	0.05	0.02	asserted 0.95

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

We apply Garnier's method to solve the Plateau problem for maximal surfaces in Minkowski 3-space. Our study relies on the improved version we gave of R. Garnier's resolution of the Plateau problem for polygonal boundary curves in Euclidean 3-space. Since in Minkowski space the method does not allow us to avoid the existence of singularities, the appropriate framework is to consider maxfaces -- generalized maximal surfaces without branch points, introduced by M. Umehara and K. Yamada. We prove that any given spacelike polygonal curve in generic position in Minkowski 3-space bounds at least one maxface of disk-type. This is a new result, since the only known result for the Plateau problem in Minkowski space (due to N. Quien) deals with boundary curves of regularity $C^{3,\alpha}$.

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2 arXiv papers cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Nevertheless, there are several advances in the mathematical literature regarding the Plateau problem in Lorentzian signature, see for example [ref] and references therein.

From Gross-Manes to Alday-Maldacena · Introduction · 0.90

What it cites · 15 sentences without a confident label

Cites as background 1 paper

See [ref] for a more general setting, and for more details on the spinor representation in Euclidean case.

The Spinor Representation of Surfaces in Space · Maximal surfaces · 0.93

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