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Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature

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Mingliang Cai, Greg Galloway

Differential Geometry · Submitted 21 Apr 1998

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

The authors' main claim The following version of a conjecture of Fischer-Colbrie and Schoen is proved: If M is a complete Riemannian 3-manifold with nonnegative scalar curvature which contains a two-sided torus S which is of least area in its isotopy class then M is flat.

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arXiv IDmath/9804096 (version 2)
Pith ID34D5KW7Z, Pith's permanent record of this paper and its review
CitationGreg Galloway, Mingliang Cai. "Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature." arXiv:math/9804096 (1998). https://pith.science/paper/34D5KW7Z
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,313 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.22 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

The following version of a conjecture of Fischer-Colbrie and Schoen is proved: If M is a complete Riemannian 3-manifold with nonnegative scalar curvature which contains a two-sided torus S which is of least area in its isotopy class then M is flat. This follows from a local version derived in the paper.

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 The following version of a conjecture of Fischer-Colbrie and Schoen is proved: If M is a complete Riemannian 3-manifold with nonnegative scalar curvature which contains a two-sided torus S which is of least area in its isotopy class then M is flat. result 0.92 0.86 0.95 asserted 1.00
s1 This follows from a local version derived in the paper. method 0.46 0.89 0.05 asserted 0.99

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

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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

The paper
Original abstract

The following version of a conjecture of Fischer-Colbrie and Schoen is proved: If M is a complete Riemannian 3-manifold with nonnegative scalar curvature which contains a two-sided torus S which is of least area in its isotopy class then M is flat. This follows from a local version derived in the paper.

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