McKean Rigidity for Cocompact Negatively Curved Manifolds and the $p$-Laplacian

Bo Zhu; Kuntao Jin

McKean Rigidity for Cocompact Negatively Curved Manifolds and the $p$-Laplacian

T0 review · 0 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that if the universal cover of a closed manifold with sectional curvature at most −1 attains McKean's sharp spectral lower bound, or any p-version of it, the cover must be hyperbolic space.

desk verdict A substantial and likely correct rigidity theorem whose only real vulnerability is the leafwise diffusion construction in §7, which needs referee scrutiny rather than desk rejection. read the letter →

arxiv 2608.00944 v1 pith:PXH4I67L submitted 2026-08-02 math.DG

Kuntao Jin , Bo Zhu This is my paper

classification math.DG MSC 53C2458J5035J9258J65

keywords p-LaplacianMcKeaninequalitybottomspectrumBusemannfunctionhorosphericalsuspensionleafwisediffusionspectralrigiditynegativecurvature

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a spectral rigidity theorem: if the universal cover $\widetilde M$ of a closed Riemannian manifold with sectional curvature at most $-1$ attains the sharp lower bound $\lambda_1(\widetilde M)=(m-1)^2/4$, then $\widetilde M$ is isometric to hyperbolic space $\mathbb H^m(-1)$. The same is proved for the variational $p$-Laplacian for every $1

What carries the argument

The central mechanism is the compact horospherical suspension $Z=(\widetilde M\times\partial_\infty\widetilde M)/\Gamma$, homeomorphic to the unit tangent bundle $SM$, with leaves $L_\xi$ covered by $\widetilde M$. The argument's hinge is the leafwise drift operator $L=\Delta_{\mathrm{leaf}}-(m-1)\langle V,\nabla_{\mathrm{leaf}}\cdot\rangle$, whose maximal continuous realization generates a conservative Feller semigroup; a stationary probability measure for $L$ has leaf-saturated support. The two McKean defects of a minimizing sequence are converted into (i) stationarity of the limiting measure under $L$ and (ii) containment of its support in $D^{-1}(0)$, the zero set of the Busemann Laplaci

What would settle it

Construct a closed Riemannian manifold with sectional curvature at most $-1$ whose universal cover is not $\mathbb H^m(-1)$ but for which a normalized minimizing sequence yields a limiting measure on the horospherical suspension whose support is not a union of complete leaves. Concretely, one could compute $\lambda_{1,p}(\widetilde M)$ for a nonhyperbolic warped-product or finite-volume example and check whether equality $((m-1)/p)^p$ holds for some $p$, which would contradict Theorem 1.2.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that McKean's spectral lower bound is rigid in the cocompact setting: if $\lambda_1(\widetilde M)=(m-1)^2/4$ or, more generally, $\lambda_{1,p}(\widetilde M)=((m-1)/p)^p$ for some $p\in(1,\infty)$, then the universal cover is isometric to $\mathbb H^m(-1)$. The proof identifies the two McKean defects—a first-order gradient defect and a Busemann Laplacian defect—and shows both vanish along a single minimizing sequence. Pushing the mass to the compact horospherical suspension $Z=(\widetilde M\times\partial_\infty\widetilde M)/\Gamma$ gives a limiting probability measure supported on the zero set of the Busemann defect, while the first defect makes it

Load-bearing premise

The load-bearing premise is the existence and positivity of a conservative leafwise heat semigroup on the compact foliated horospherical suspension, whose leaves may be noncompact and nonproper; if that semigroup failed for the $C^{\infty,0}$ coefficients used here, the support of the limiting measure could fail to be leaf-saturated and the rigidity conclusion would not follow.

Editorial extensions

If this is right

If the paper is correct, for any closed manifold with sectional curvature at most $-1$, spectral equality $\lambda_1(\widetilde M)=((m-1)/2)^2$ detects hyperbolicity without any volume-growth or entropy input.
Equality in the $p$-Laplacian bound for any single $p\in(1,\infty)$ already forces hyperbolicity; once hyperbolicity holds, equality holds for every $p$.
The stationary-measure/leaf-saturation method gives a template for other spectral rigidity problems where minimizing-sequence defects can be localized on a compact foliated suspension.
The paper's examples show that both a uniform lower curvature bound and compactness of the suspension are genuinely needed; removing either produces nonhyperbolic covers that still attain the sharp spectral value.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the same defect-to-stationarity mechanism may extend to other leafwise operators with the same drift coefficient $m-1$, for example certain nonlinear or magnetic Laplacians, whenever a convexity-defect identity and a leafwise heat semigroup are available.
The finite-volume example in Section 6.4 suggests that without cocompactness, equality in the sharp bound can coexist with nonhyperbolicity because minimizing mass escapes to a cusp; this hints that suitable tightness conditions, rather than curvature bounds alone, might be the right replacement for compactness.
Problem 6.5 asks whether volume-entropy rigidity extends to finite-volume quotients; a positive answer would unify the spectral and entropy routes, while a negative one would sharpen the boundary between them.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a real theorem, not a repackaging. The p-Laplacian rigidity is new, and the quadratic case under sec ≤ −1 uses a defect-plus-stationary-measure argument genuinely different from Wang's Ricci-based proof. The McKean square identity and its p-analogue are clean, and the reduction to a linear transport equation for the density u_j^p is a nice trick.

The achievements: Theorem 1.2 is exactly the kind of sharp rigidity result that belongs in the literature. The proof is honest about where cocompactness enters, and the counterexamples in §6.2 usefully separate the two roles of closedness. The exact identities are there, no fitted parameters, and the final step from ΔB = m−1 to hyperbolicity is standard and clean.

The soft spot is Section 7. The whole rigidity conclusion hangs on Lemma 5.2(iii) — leaf saturation of stationary supports — which depends on Theorem 7.5(iii), the identification of the global Feller semigroup with the intrinsic heat semigroup on each leaf. The duplicated-leaf construction is the right way to handle nonproper leaves, but the verification of the foliated structure and coefficient regularity on the enlarged space is terse. The stress-test note I received claims the drift coefficient becomes discontinuous when a nonproper leaf accumulates on another leaf. I don't think that's right as stated: the drift is the pullback of V, and V is continuous on the suspension, so the values on the copied leaf converge correctly. What genuinely needs checking is the C^{1,0} regularity of the leafwise derivatives of the drift across the duplicated-leaf boundary, and more generally that the enlarged space fits the hypotheses of Theorem 7.5(i)–(ii). Candel's paper is cited for this, but the paper's summary is thin enough that a referee should ask for details.

If that gap is repairable — and I expect it is — the rest of the proof is solid. The citation pattern looks fine; self-citations are to companion papers and are not padded.

Bottom line: this deserves peer review. It's a paper for spectral geometers and rigidity people. I'd want the referee to focus on §7.3 and the duplicated-leaf construction. My own verdict would be accept-with-revisions once the Section 7 details are filled in.

Referee Report

0 major / 3 minor

Summary. The paper proves McKean rigidity for universal covers of closed Riemannian manifolds with sec ≤ -1. Theorem 1.1 states that λ_1(\tilde M) = (m-1)^2/4 iff \tilde M is isometric to hyperbolic space H^m(-1). Theorem 1.2 extends this to the variational p-Laplacian: λ_{1,p}(\tilde M) ≥ ((m-1)/p)^p, and equality for some p holds iff \tilde M is hyperbolic. The proof introduces an exact square identity (Lemma 2.2) that separates the spectral excess into a first-order defect and a Busemann Laplacian defect. For a normalized minimizing sequence, both defects vanish. The measures u_j^2 dV are pushed forward to the compact horospherical suspension Z = (\tilde M × ∂_∞ \tilde M)/Γ ≅ SM. A weak limit μ is supported in the zero set of the Busemann defect D. The first defect yields stationarity of μ for the leafwise drift operator L = Δ_leaf - (m-1)⟨V, ∇_leaf ·⟩. Using Candel's leafwise diffusion theory developed in Section 7, the support is shown to be leaf-saturated, so one complete leaf in supp μ gives a Busemann function with ΔB = m-1 everywhere. Busemann Hessian comparison and a warped-product splitting then force hyperbolicity. The p-case uses a Bregman-divergence identity (Lemma A.2) to obtain the same L^1 transport equation with coefficient m-1, so the same diffusion and support arguments apply. The paper also contains examples (§6.2) showing why cocompactness and a lower curvature bound are needed.

Significance. The result is a significant spectral rigidity theorem in negatively curved Riemannian geometry. It identifies hyperbolic space as the unique extremal universal cover among closed manifolds with sec ≤ -1, both for the classical Laplacian and for all p-Laplacians. The proof is elegant and surprisingly soft: it converts the McKean defect into a stationary measure on the compact horospherical suspension, and then uses heat-kernel positivity to propagate a single zero-defect point along a complete leaf. The paper gives parameter-free, exact identities, and it develops a substantial amount of leafwise diffusion theory (Section 7), including a careful treatment of nonproper leaves via Candel's duplicated-leaf construction. The p-Laplacian extension is nontrivial and the equality case is handled uniformly in p. The examples in §6.2 clarify the roles of compactness and two-sided curvature. If correct, the paper answers a natural rigidity question and will be of interest to geometers and spectral theorists.

minor comments (3)

[§7.3, Step 1 (foliated structure of \bar X_L)] The construction of the foliated atlas on the duplicated-leaf space is only sketched. In particular, the sentence 'topologize their transverse parameters by (7.15)' needs to be expanded. A skeptical reader worries that when a nonproper leaf accumulates on a different leaf, the drift coefficient of \bar A could be discontinuous. The resolution is that the copied plaques in a chart are assigned the original transverse coordinates (possibly reanchored by the deck transformation), so the coefficients of \bar A are the same continuous functions of (y,τ) as on the original plaques. Please state this explicitly and verify that the resulting atlas is C^{r,0} with \bar A having C^{1,0} first-order coefficients. This is a local clarity issue; the construction itself appears sound.
[§7.3, Step 1 (Hausdorff property)] In the Hausdorff check, the basis element is written as N(X,V), but V is not compact; the definition of N(U,K) requires K compact. The authors presumably mean N(X,\overline{V}) or a compact neighborhood contained in V. This is a minor notational slip.
[§5.4 / §6.1] The transition from a single Busemann function with ΔB = m-1 to the warped-product splitting is clean, but it would help to add a sentence explicitly noting that the lower curvature bound in Lemma 6.1 is used only to rule out nontrivial sectional curvature of the horosphere H, while the upper bound gives the Hessian identity. This is implicit but could be stated for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rigidity proof is an exact defect identity plus an externally sourced leafwise semigroup construction, with no fitted parameters and no load-bearing self-citation.

full rationale

The paper's central derivation is self-contained and non-circular. The McKean square identity (Lemma 2.2) and its p-version (Lemma A.2) are exact integration-by-parts identities; the lower bound follows from the pointwise nonnegativity of the two defect terms under ΔB ≥ m−1 (Proposition 2.1). Equality forces both defects to vanish along a minimizing sequence (Proposition 2.3 / A.3), which is a direct consequence of the identity and the variational definition, not a fitted assumption. The passage to the compact horospherical suspension (Section 3) is a change of variables; the limiting measure μ satisfies supp μ ⊆ D^{-1}(0) by continuity (Proposition 4.1). The first defect gives ∫ Lf dμ = 0 for the leafwise drift operator (Proposition 5.3), and the leafwise Feller semigroup is constructed in Section 7 following Candel's external theorem, not imported from the authors' own prior work. Leaf saturation of stationary supports (Lemma 5.2(iii), Proposition 7.9) is derived from strict positivity of the leafwise heat kernel, not assumed. The final Riemannian step (Lemma 6.1) is a standard Hessian-comparison and warped-product argument. The only self-citations ([WZ24], [WZ26], same author Bo Zhu) appear in the related-work section and are not used in the proof. No parameter is fitted, no known theorem is renamed as a prediction, and no uniqueness result is imported from the authors' own prior work. The possible concern that the duplicated-leaf construction in Theorem 7.5(iii) may fail to have C^{1,0} drift coefficients is a mathematical gap in the proof of a lemma, not a circularity: the conclusion is not assumed or encoded in the hypotheses.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof rests on standard Riemannian comparison and on a substantial foliated diffusion existence theorem; no parameters are fitted to data.

assumptions (4)

domain assumption The leafwise diffusion semigroup on the compact C^{∞,0} foliated space Z is conservative, Feller, and has strictly positive smooth heat kernels on every leaf (Theorem 7.5, following Candel [Can03]).
Used in Lemma 5.2 to turn the first-defect integral into stationarity and then leaf saturation; the entire support argument depends on it.
standard math Under -A^2 ≤ sec ≤ -1, Busemann functions are C^2 and satisfy Hessian and Laplacian comparison (Proposition 2.1 via stable Jacobi fields).
Provides the defect identity and the key inequality ΔB ≥ m-1.
standard math The horospherical suspension (M̃ × ∂∞ M̃)/Γ is a compact metrizable C^{∞,0} foliated space with Riemannian leaves, homeomorphic to the unit tangent bundle SM.
Compactness prevents loss of mass; the leafwise structure carries the Busemann data.
standard math Strict uniform convexity of the p-norm and the Bregman divergence estimates (Lemma A.1).
Yields strong Lp control of the first defect and the L1 transport equation for ρ = u^p.

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Cite this review

Pith. "Pith review of McKean Rigidity for Cocompact Negatively Curved Manifolds and the $p$-Laplacian." pith.science (2026). https://pith.science/paper/PXH4I67L

@misc{pith2026260800944,
  author       = {Pith},
  title        = {Pith review of: McKean Rigidity for Cocompact Negatively Curved Manifolds and the \(p\)-Laplacian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PXH4I67L}},
  note         = {Machine review of arXiv:2608.00944}
}

abstract

Let $(M^m,g)$ be a closed Riemannian manifold with $\sec_g\leq-1$. We prove that the bottom spectrum of its universal cover attains McKean's lower bound if and only if the universal cover is hyperbolic space of constant sectional curvature $-1$. More generally, for every $1<p<\infty$, the variational $p$-fundamental tone satisfies \[ \lambda_{1,p}(\wti M) \geq\left(\frac{m-1}{p}\right)^p, \] and equality for some $p\in(1,\infty)$ holds if and only if $\wti M\cong\bH^m(-1)$. In that case, equality holds for every $p\in(1,\infty)$. The proof converts the two McKean defects of a minimizing sequence into a stationary probability measure on the compact horospherical suspension; heat-kernel positivity then forces its zero-defect support to contain a complete leaf.

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [8]

MR4319403 [Che70] Jeff Cheeger,A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis, Princeton University Press, Princeton, NJ, 1970, pp. 195–199. MR0402831 [Che75] Shiu Yuen Cheng,Eigenvalue comparison theorems and its geometric applications, Math. Z.143(1975), no. 3, 289–297. MR0378001 [DPPS09] Fran¸ coise Dal’Bo, Marc Peign´...

work page 1970

[2] [41]

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MR4736650 [MW26] ,Bottom spectrum and parabolicity of 3-manifolds with scalar curvature lower bound (2026), Preprint, available athttps://arxiv.org/abs/2607.06508. [Pol14] Guillaume Poliquin,Bounds on the principal frequency of thep-Laplacian, Geometric and spectral analysis, Contemp. Math., vol. 630, Amer. Math. Soc., Providence, RI, 2014, pp. 349–366. M...

work page Pith review arXiv 2026

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MR0224110 [BMZ18] Thomas Barthelm´ e, Ludovic Marquis, and Andrew Zimmer,Entropy rigidity of Hilbert and Riemannian metrics, Int

English translation. MR0224110 [BMZ18] Thomas Barthelm´ e, Ludovic Marquis, and Andrew Zimmer,Entropy rigidity of Hilbert and Riemannian metrics, Int. Math. Res. Not. IMRN22(2017), 6841–6866. MR3737323 [BCH25a] G´ erard Besson, Gilles Courtois, and Sa’ar Hersonsky,Rigidity of flat holonomies, Ergodic Theory Dynam. Systems45(2025), no. 4, 1048–1077. MR4874...

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MR0501173 [Led10] Fran¸ cois Ledrappier,Linear drift and entropy for regular covers, Geom. Funct. Anal.20 (2010), no. 3, 710–725. MR2720229 [L W10] Fran¸ cois Ledrappier and Xiaodong Wang,An integral formula for the volume entropy with applications to rigidity, J. Differential Geom.85(2010), no. 3, 461–477. MR2739810 [Li12] Peter Li,Geometric analysis, Ca...

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MR1814364 56 KUNTAO JIN AND BO ZHU [G¨ u60] Paul G¨ unther,Einige S¨ atze ¨ uber das Volumenelement eines Riemannschen Raumes, Publ

Reprint of the 1998 edition. MR1814364 56 KUNTAO JIN AND BO ZHU [G¨ u60] Paul G¨ unther,Einige S¨ atze ¨ uber das Volumenelement eines Riemannschen Raumes, Publ. Math. Debrecen7(1960), no. 1–4, 78–93. MR0141058 [HIH77] Ernst Heintze and Hans-Christoph Im Hof,Geometry of horospheres, J. Differential Geom. 12(1977), no. 4, 481–491. MR0512919 [HPS77] Morris ...

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Bowditch,Cut points and canonical splittings of hyperbolic groups, Acta Math

MR2267655 [Bow98] Brian H. Bowditch,Cut points and canonical splittings of hyperbolic groups, Acta Math. 180(1998), no. 2, 145–186. MR1638764 [Bre11] Haim Brezis,Functional analysis, Sobolev spaces and partial differential equations, Univer- sitext, Springer, New York,

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[7] [2011]

Z.178 (1981), no

MR2759829 [Bro81] Robert Brooks,A relation between growth and the spectrum of the Laplacian, Math. Z.178 (1981), no. 4, 501–508. MR0638814 [Can03] Alberto Candel,The harmonic measures of Lucy Garnett, Adv. Math.176(2003), no. 2, 187–247. MR1982882 [CC22] Francisco G. de S. Carvalho and Marcos P. de A. Cavalcante,On the fundamental tone of thep-Laplacian o...

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Differential Geom

MR2962229 [L W01] Peter Li and Jiaping Wang,Complete manifolds with positive spectrum, J. Differential Geom. 58(2001), no. 3, 501–534. MR1906784 [L W02] ,Complete manifolds with positive spectrum. II, J. Differential Geom.62(2002), no. 1, 143–162. MR1987380 [Liu11] Gang Liu,A short proof to the rigidity of volume entropy, Math. Res. Lett.18(2011), no. 1, ...

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