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Existence of infinitely many minimal hypersurfaces in low dimensions, after F.C. Marques, A.A. Neves et A. Song (Bourbaki Seminar)

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Tristan Rivière

Differential Geometry · Submitted 17 May 2019

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

The authors' main claim The goal of this presentation is to prove that this rigidity result does not extend to higher dimensions and, more precisely, on any smooth closed riemannian manifod of arbitrary dimension between 3 and 7 there exists infinitely many closed embedded minimal surfaces.

how it is stated · asserted 1.00 read the record →

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arXiv ID1905.07120 (version 1)
Pith IDWIJP4GXN, Pith's permanent record of this paper and its review
CitationTristan Rivi\`ere. "Existence of infinitely many minimal hypersurfaces in low dimensions, after F.C. Marques, A.A. Neves et A. Song (Bourbaki Seminar)." arXiv:1905.07120 (2019). https://pith.science/paper/WIJP4GXN
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,010 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.29 · human participants 0.02 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.74 · compares with prior results 0.93 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

A classical result by Marston Morse asserts that on some ellipsoids of ${\mathbb R}^3$ there exists exactly 3 closed and simple geodesics. The goal of this presentation is to prove that this rigidity result does not extend to higher dimensions and, more precisely, on any smooth closed riemannian manifod of arbitrary dimension between 3 and 7 there exists infinitely many closed embedded minimal surfaces. We are going to present the origins of this theorem as well as it's proof given recently by Antoine Song.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 A classical result by Marston Morse asserts that on some ellipsoids of ${\mathbb R}^3$ there exists exactly 3 closed and simple geodesics. background 1.00 0.02 0.02 asserted 1.00
s1 The goal of this presentation is to prove that this rigidity result does not extend to higher dimensions and, more precisely, on any smooth closed riemannian manifod of arbitrary dimension between 3 and 7 there exists infinitely many closed embedded minimal surfaces. result 0.64 0.54 0.93 asserted 1.00
s2 We are going to present the origins of this theorem as well as it's proof given recently by Antoine Song. contribution 0.69 0.23 0.04 asserted 0.83

What kind of paper the classifier reports this is: theoretical result 0.79 · evidence from proof or derivation 0.96.

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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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

The paper
Original abstract

A classical result by Marston Morse asserts that on some ellipsoids of ${\mathbb R}^3$ there exists exactly 3 closed and simple geodesics. The goal of this presentation is to prove that this rigidity result does not extend to higher dimensions and, more precisely, on any smooth closed riemannian manifod of arbitrary dimension between 3 and 7 there exists infinitely many closed embedded minimal surfaces. We are going to present the origins of this theorem as well as it's proof given recently by Antoine Song.

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What this paper does with the arXiv papers it cites

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

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

What it cites · 6 sentences without a confident label

Cites as background 1 paper
  • Un résultat très intéressant de Marques, Neves et Song [ref] affirme que dans le cas de métriques génériques il existe une famille dénombrable d'hypersurfaces minimales plongées et fermées satisfaisant une propriété d'équidistribution.

    Equidistribution of minimal hypersurfaces for generic metrics · Conclusion · 0.84

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