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Constant scalar curvature metrics with isolated singularities

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Rafe Mazzeo, Frank Pacard

dg-ga · Submitted 14 May 1996

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

The authors' main claim We extend the results and methods of \cite{MP} to prove the existence of constant positive scalar curvature metrics $g$ which are complete and conformal to the standard metric on $S^N \setminus \Lambda$, where $\Lambda$ is a disjoint union of submanifolds of dimensions between 0 and $(N-2)/2$.

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arXiv IDdg-ga/9605004 (version 1)
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CitationFrank Pacard, Rafe Mazzeo. "Constant scalar curvature metrics with isolated singularities." arXiv:dg-ga/9605004 (1996). https://pith.science/paper/DYUPXGVQ
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The compiled record

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,183 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.97 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We extend the results and methods of \cite{MP} to prove the existence of constant positive scalar curvature metrics $g$ which are complete and conformal to the standard metric on $S^N \setminus \Lambda$, where $\Lambda$ is a disjoint union of submanifolds of dimensions between 0 and $(N-2)/2$. The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by Schoen \cite{S}, but the proof we give here, based on the techniques of \cite{MP}, is more direct, and provides more information about their geometry. When $\Lambda$ is discrete we also establish that these solutions are smooth points in the moduli spaces of all such solutions introduced and studied in \cite{MPU1} and \cite{MPU2}

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 We extend the results and methods of \cite{MP} to prove the existence of constant positive scalar curvature metrics $g$ which are complete and conformal to the standard metric on $S^N \setminus \Lambda$, where $\Lambda$ is a disjoint union of submanifolds of dimensions between 0 and $(N-2)/2$. contribution 0.54 0.95 0.87 asserted 1.00
s1 The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by Schoen \cite{S}, but the proof we give here, based on the techniques of \cite{MP}, is more direct, and provides more information about their geometry. contribution 0.80 0.86 0.13 asserted 1.00
s2 When $\Lambda$ is discrete we also establish that these solutions are smooth points in the moduli spaces of all such solutions introduced and studied in \cite{MPU1} and \cite{MPU2} result 0.93 0.93 0.10 asserted 1.00

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

We extend the results and methods of \cite{MP} to prove the existence of constant positive scalar curvature metrics $g$ which are complete and conformal to the standard metric on $S^N \setminus \Lambda$, where $\Lambda$ is a disjoint union of submanifolds of dimensions between 0 and $(N-2)/2$. The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by Schoen \cite{S}, but the proof we give here, based on the techniques of \cite{MP}, is more direct, and provides more information about their geometry. When $\Lambda$ is discrete we also establish that these solutions are smooth points in the moduli spaces of all such solutions introduced and studied in \cite{MPU1} and \cite{MPU2}

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