pith. sign in

arxiv: 2309.09166 · v5 · submitted 2023-09-17 · 🧮 math.DG

Rotational symmetry of complete shrinking gradient Yamabe solitons

Shun Maeta This is my paper

Pith reviewed 2026-05-24 07:12 UTC · model grok-4.3

classification 🧮 math.DG
keywords Yamabe solitonsgradient solitonsshrinking solitonsrotational symmetryscalar curvaturecomplete Riemannian manifoldsPerelman's conjecture
0
0 comments X

The pith

Complete shrinking gradient Yamabe solitons are rotationally symmetric under the scalar curvature condition

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that any nontrivial complete shrinking gradient Yamabe soliton is rotationally symmetric if its scalar curvature is at least the soliton constant at every point and strictly greater at some point. This condition is shown to be optimal in higher dimensions. The result settles the Yamabe-soliton analogue of Perelman's conjecture, which concerns the symmetry properties of these self-similar solutions to the Yamabe flow.

Core claim

Any nontrivial complete shrinking gradient Yamabe soliton whose scalar curvature is bounded below by the soliton constant everywhere and is strictly greater than the constant at some point is rotationally symmetric. This assumption is optimal for higher dimensions.

What carries the argument

The scalar curvature condition on the shrinking gradient Yamabe soliton, which is used to prove that the manifold must be rotationally symmetric.

If this is right

  • The soliton must have the geometry of a sphere or a rotationally symmetric space.
  • The result applies to all dimensions with optimality noted in dimensions greater than two.
  • It provides a classification for these solitons under the given curvature assumption.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The optimality of the assumption suggests that dropping the strict inequality allows for non-symmetric examples.
  • Similar techniques might apply to other geometric solitons like Ricci solitons.
  • Explicit constructions in low dimensions could verify the necessity of the strict inequality.

Load-bearing premise

The scalar curvature is bounded below by the soliton constant everywhere and strictly exceeds it at some point.

What would settle it

Discovery of a nontrivial complete shrinking gradient Yamabe soliton that is not rotationally symmetric while satisfying the scalar curvature condition would disprove the claim.

read the original abstract

In this paper, we show that any nontrivial complete shrinking gradient Yamabe soliton whose scalar curvature is bounded below by the soliton constant everywhere and is strictly greater than the constant at some point is rotationally symmetric. This assumption is optimal for higher dimensions. This result resolves the Yamabe-soliton analogue of Perelman's conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The paper proves that any nontrivial complete shrinking gradient Yamabe soliton whose scalar curvature is bounded below by the soliton constant everywhere and strictly exceeds it at some point is rotationally symmetric. The assumption is claimed to be optimal in higher dimensions, and the result is presented as resolving the Yamabe-soliton analogue of Perelman's conjecture.

Significance. If the proof is correct, the result is significant for geometric analysis as it establishes a rotational symmetry theorem for shrinking gradient Yamabe solitons under a natural and sharp scalar-curvature condition. This provides a classification-type statement analogous to known results for Ricci solitons and contributes to understanding the Yamabe flow. The paper supplies a mathematical proof of the stated theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report and summary of the manuscript. No specific major comments appear under the MAJOR COMMENTS section of the report. We therefore have no individual points to address. The 'uncertain' recommendation may reflect a desire for further verification of the proof; we remain available to supply additional details or clarifications if requested by the editor or referee.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract states a theorem on rotational symmetry of shrinking gradient Yamabe solitons under a scalar curvature lower bound with strict inequality at one point, resolving an analogue of Perelman's conjecture. No derivation chain, equations, self-citations, or fitted parameters are visible in the given text. The result is presented as a direct proof under stated assumptions, with no reduction of outputs to inputs by construction or load-bearing self-citation. This is consistent with a self-contained pure-mathematics argument; score 0 is the appropriate default when no circular steps can be exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.0 · 5558 in / 988 out tokens · 21738 ms · 2026-05-24T07:12:00.484367+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Classification of low-dimensional complete gradient Yamabe solitons

    math.DG 2024-05 unverdicted novelty 6.0

    Complete classification of nontrivial non-flat two- and three-dimensional complete gradient Yamabe solitons.

  2. Triviality, Rotational Symmetry, and Classification of Complete Expanding Gradient Yamabe Solitons

    math.DG 2024-12 unverdicted novelty 5.0

    Complete classification of nontrivial complete expanding gradient Yamabe solitons when scalar curvature exceeds or falls below the soliton constant.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · cited by 2 Pith papers

  1. [1]

    Brendle, Convergence of the Yamabe flow for arbitrary initial energy , J

    S. Brendle, Convergence of the Yamabe flow for arbitrary initial energy , J. Differential Geom. (2005) 69, 217–278

    work page 2005

  2. [2]

    Brendle, Convergence of the Yamabe flow in dimension 6 and higher , Invent

    S. Brendle, Convergence of the Yamabe flow in dimension 6 and higher , Invent. Math. (2007) 170, 541–576

    work page 2007

  3. [3]

    Brendle, Rotational symmetry of self-similar solutions to the Ricci flow, In- vent

    S. Brendle, Rotational symmetry of self-similar solutions to the Ricci flow, In- vent. Math. (2013) 194, 731–764

    work page 2013

  4. [4]

    Brendle, Rotational symmetry of Ricci solitons in higher dimensions , J

    S. Brendle, Rotational symmetry of Ricci solitons in higher dimensions , J. Dif- ferential Geom. (2014) 97, 191–214

    work page 2014

  5. [5]

    H.-D. Cao, X. Sun and Y. Zhang, On the structure of gradient Yamabe solitons, Math. Res. Lett. (2012) 19, 767–774

    work page 2012

  6. [6]

    Catino, C

    G. Catino, C. Mantegazza and L. Mazzieri, On the global structure of confor- mal gradient solitons with nonnegative Ricci tensor , Commun. Contemp. Math. (2012) 14, 12pp

    work page 2012

  7. [7]

    Cheeger and T

    J. Cheeger and T. H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (1996) 144, 189–237

    work page 1996

  8. [8]

    Daskalopoulos and N

    P. Daskalopoulos and N. Sesum, The classification of locally conformally flat Yamabe solitons, Adv. Math. (2013) 240, 346–369

    work page 2013

  9. [9]

    Hamilton, Lectures on geometric flows , (1989), unpublished

    R. Hamilton, Lectures on geometric flows , (1989), unpublished. ROTATIONAL SYMMETRY OF STEADY GRADIENT YAMABE SOLITONS 5

    work page 1989

  10. [10]

    S. Y. Hsu, A note on compact gradient Yamabe solitons, J. Math. Anal. Appl. (2012) 388 (2), 725–726

    work page 2012

  11. [11]

    Maeta, Classification of gradient conformal solitons, arXiv:2107.05487[math DG]

    S. Maeta, Classification of gradient conformal solitons, arXiv:2107.05487[math DG]

    work page arXiv

  12. [12]

    Perelman, The entropy formula for the Ricci flow and its geometric appli - cations, arXiv math.DG/0211159, (2002)

    G. Perelman, The entropy formula for the Ricci flow and its geometric appli - cations, arXiv math.DG/0211159, (2002)

    work page arXiv 2002

  13. [13]

    Petersen, Riemannian Geometry

    P. Petersen, Riemannian Geometry. Third edition, Graduate Texts in Mathe- matics, 171. Springer (2016)

    work page 2016

  14. [14]

    Tashiro, Complete Riemannian manifolds and some vector fields , Trans

    Y. Tashiro, Complete Riemannian manifolds and some vector fields , Trans. Amer. math. Soc., 117 (1965), 251–275. Department of Mathematics, F aculty of Education, and Department of Mathematics and Informatics, Graduate School of Science and En- gineering, Chiba University, 1-33, Yayoicho, Inage, Chiba , 263-8522, Japan. Email address : shun.maeta@gmail.com...

    work page 1965