Scalar curvature, sharp bottom spectrum and geometric rigidity

Bo Zhu; Jinmin Wang

arxiv: 2606.11957 · v1 · pith:RCC3SLO5new · submitted 2026-06-10 · 🧮 math.DG

Scalar curvature, sharp bottom spectrum and geometric rigidity

Jinmin Wang , Bo Zhu This is my paper

Pith reviewed 2026-06-27 08:31 UTC · model grok-4.3

classification 🧮 math.DG

keywords scalar curvaturebottom spectrumgeometric rigidityhyperbolic manifoldsLaplacianuniversal covernonpositive sectional curvature

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The pith

Closed manifolds with scalar curvature at least -n(n-1) and equality in the sharp bottom spectrum must be hyperbolic under topological assumptions.

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

The paper establishes rigidity for the equality case in a sharp bottom spectrum estimate when a scalar curvature lower bound holds. It shows that a closed manifold satisfying Sc_g ≥ -n(n-1) together with λ₁ of the universal cover equal to (n-1)²/4 must be hyperbolic, provided the topological assumptions from prior work are met. The result applies directly to closed hyperbolic manifolds and to closed manifolds that admit a metric of nonpositive sectional curvature. A reader would care because the combination of curvature and spectral data then forces the manifold to have constant negative curvature geometry.

Core claim

We prove rigidity in the equality case of the sharp bottom spectrum estimate under scalar curvature lower bound. Under the same topological assumptions as in our previous work, a closed manifold (M,g) with Sc_g ≥ -n(n-1) and λ₁(̃M,̃g)=(n-1)²/4 must be hyperbolic. This gives rigidity results for closed hyperbolic manifolds and for closed manifolds admitting a metric of nonpositive sectional curvature.

What carries the argument

The equality case λ₁(̃M,̃g) = (n-1)²/4 of the bottom spectrum on the universal cover, which combines with the scalar curvature bound Sc_g ≥ -n(n-1) to force the manifold to be hyperbolic.

If this is right

Closed hyperbolic manifolds satisfy the rigidity conclusion under the stated conditions.
Closed manifolds that admit a metric of nonpositive sectional curvature are rigid in the same sense.
The equality case of the bottom spectrum estimate forces hyperbolicity when the scalar curvature bound holds.

Where Pith is reading between the lines

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

The result may supply a spectral criterion for detecting hyperbolicity in manifolds already known to satisfy negative curvature bounds.
Similar rigidity statements could be tested on manifolds whose fundamental groups satisfy the same topological conditions but whose curvature is only controlled in an average sense.
The technique might extend to rigidity questions involving other operators on the universal cover beyond the Laplacian.

Load-bearing premise

The topological assumptions inherited from the authors' previous work are needed for the conclusion that the manifold must be hyperbolic.

What would settle it

A closed manifold satisfying Sc_g ≥ -n(n-1), λ₁(̃M,̃g) exactly equal to (n-1)²/4, and the topological assumptions, yet failing to be hyperbolic, would falsify the claim.

read the original abstract

We prove rigidity in the equality case of the sharp bottom spectrum estimate under scalar curvature lower bound. Under the same topological assumptions as in our previous work, a closed manifold $(M,g)$ with $\mathrm{Sc}_g\geq -n(n-1)$ and $\lambda_1(\widetilde M,\widetilde g)=(n-1)^2/4$ must be hyperbolic. This gives rigidity results for closed hyperbolic manifolds and for closed manifolds admitting a metric of nonpositive sectional curvature.

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.

Desk Editor's Note private letter to a colleague

This paper adds a scalar curvature lower bound to the authors' prior spectral estimate and claims rigidity to hyperbolicity in the equality case, but only under topological assumptions left implicit from their earlier work.

read the letter

The main takeaway is that under the same topological assumptions as the authors' previous spectrum paper, a closed manifold with scalar curvature at least -n(n-1) and equality in the bottom spectrum of the universal cover must be hyperbolic. This is positioned as giving rigidity for closed hyperbolic manifolds and for closed manifolds that admit a metric of nonpositive sectional curvature.

What is new is the combination of the scalar curvature bound with the spectral equality condition to reach the rigidity conclusion. Their earlier work apparently handled the sharp bottom spectrum estimate, and this extends it by showing that equality plus the curvature lower bound forces the geometry to be hyperbolic. That step is a logical continuation rather than a complete overhaul.

The paper states the claim cleanly and points to the relevant applications in the abstract. If the full proof follows the same lines as the prior spectral results and the derivations check out, it organizes some existing phenomena in a useful way for people working on manifolds with nonpositive curvature.

The soft spot is the dependence on topological assumptions that are not restated or characterized here. The abstract simply refers to "the same topological assumptions as in our previous work" without summarizing their content or minimality. This forces any reader to consult the earlier paper to understand the precise scope or to check whether those assumptions mesh cleanly with the curvature and spectral data. It is a minor but real drawback for self-contained reading.

This is for specialists already tracking the authors' series on spectral estimates in differential geometry. A reader who has the prior paper in hand will see the extension most clearly. It is coherent enough on its own terms to merit a serious referee, even if the assumptions need to be made more explicit.

Referee Report

1 major / 1 minor

Summary. The manuscript proves a rigidity theorem for closed Riemannian manifolds: under the same topological assumptions as the authors' previous work, if Sc_g ≥ -n(n-1) and the bottom of the spectrum λ₁(̃M, ̃g) equals (n-1)²/4, then (M,g) is hyperbolic. The result is applied to obtain rigidity for closed hyperbolic manifolds and for closed manifolds admitting a metric of nonpositive sectional curvature.

Significance. If correct, the result links a scalar curvature lower bound to spectral rigidity on the universal cover, yielding geometric conclusions about hyperbolicity. The extension to manifolds with nonpositive sectional curvature broadens the scope beyond constant-curvature cases and may connect to existing rigidity theorems in geometric analysis.

major comments (1)

[Introduction] Introduction (or the paragraph stating the main theorem): The topological assumptions are referred to only as 'the same topological assumptions as in our previous work' without being restated or characterized. Because the rigidity conclusion is conditional on these hypotheses, their explicit formulation is required to determine the precise scope of the theorem and to verify that the analytic conditions (Sc_g ≥ -n(n-1) and λ₁ = (n-1)²/4) together with the assumptions indeed force hyperbolicity.

minor comments (1)

[Abstract] Abstract: The notation (̃M, ̃g) for the universal cover is used without a preceding definition; while common in the field, a brief clarification would improve readability for a broader audience.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment on improving the clarity of the main result. We address the point below.

read point-by-point responses

Referee: [Introduction] Introduction (or the paragraph stating the main theorem): The topological assumptions are referred to only as 'the same topological assumptions as in our previous work' without being restated or characterized. Because the rigidity conclusion is conditional on these hypotheses, their explicit formulation is required to determine the precise scope of the theorem and to verify that the analytic conditions (Sc_g ≥ -n(n-1) and λ₁ = (n-1)²/4) together with the assumptions indeed force hyperbolicity.

Authors: We agree that restating the topological assumptions explicitly will make the scope of the theorem clearer and easier to verify. In the revised version we will include a precise formulation of these assumptions (as stated in our previous work) directly in the introduction and in the statement of the main theorem. revision: yes

Circularity Check

1 steps flagged

Rigidity conclusion rests on unspecified topological assumptions from authors' prior work

specific steps

self citation load bearing [Abstract]
"Under the same topological assumptions as in our previous work, a closed manifold (M,g) with Sc_g ≥ -n(n-1) and λ₁(̃M,̃g)=(n-1)²/4 must be hyperbolic."

The load-bearing rigidity claim (hyperbolicity from the given curvature and spectral equality) is asserted only under topological hypotheses imported from the authors' own prior paper; those hypotheses are neither restated nor independently verified in the present work, so the central geometric conclusion reduces to the content of the self-citation.

full rationale

The paper's central rigidity statement is explicitly conditioned on 'the same topological assumptions as in our previous work' without restating or deriving them here. This creates a self-citation load-bearing dependency for the equality-case conclusion, though the analytic conditions (scalar curvature bound and bottom spectrum equality) appear to be handled independently within the current manuscript. No reduction of fitted parameters or self-definitional equations is exhibited in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are mentioned or can be extracted.

pith-pipeline@v0.9.1-grok · 5591 in / 1083 out tokens · 27495 ms · 2026-06-27T08:31:44.457200+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 1 linked inside Pith

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MR 3410545

Jean-Pierre Bourguignon, Oussama Hijazi, Jean-Louis Milhorat, Andrei Moroianu, and Sergiu Moroianu,A spinorial approach to Riemannian and conformal geometry, EMS Monographs in Mathematics, European Mathematical Society (EMS), Z¨ urich, 2015. MR 3410545

2015

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Z.143 (1975), no

Shiu Yuen Cheng,Eigenvalue comparison theorems and its geometric applications, Math. Z.143 (1975), no. 3, 289–297. MR 378001

1975

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Demetre Kazaras, Antoine Song, and Kai Xu,Scalar curvature and volume entropy of hyperbolic 3-manifolds, J. Eur. Math. Soc. (2025), arXiv:2312.00138

arXiv 2025

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Differential Geom.85(2010), no

Fran¸cois Ledrappier and Xiaodong Wang,An integral formula for the volume entropy with appli- cations to rigidity, J. Differential Geom.85(2010), no. 3, 461–477. MR 2739810

2010

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134, Cambridge University Press, Cambridge, 2012

Peter Li,Geometric analysis, Cambridge Studies in Advanced Mathematics, vol. 134, Cambridge University Press, Cambridge, 2012. MR 2962229

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2001

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Gang Liu,A short proof to the rigidity of volume entropy, Math. Res. Lett.18(2011), no. 1, 151–153. MR 2756006 SCALAR CUR V ATURE, SHARP BOTTOM SPECTRUM AND GEOMETRIC RIGIDITY 19

2011

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Ovidiu Munteanu and Jiaping Wang,Bottom spectrum of three-dimensional manifolds with scalar curvature lower bound, J. Funct. Anal.287(2024), no. 2, Paper No. 110457, 41. MR 4736650

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Xiaodong Wang,Harmonic functions, entropy, and a characterization of the hyperbolic space, J. Geom. Anal.18(2008), no. 1, 272–284. MR 2365675 (Jinmin Wang)Institute of Mathematics, Chinese Academy of Sciences Email address:jinmin@amss.ac.cn (Bo Zhu)Yau Mathematical Sciences Center, Tsinghua University Email address:zhub@tsinghua.edu.cn

2008