A Willmore-type inequality for hypersurfaces with asymptotic or integral Ricci curvature bounds

Jihye Lee

REVIEW 2 major objections 4 minor 22 references

A Willmore-type inequality for hypersurfaces with asymptotic or integral Ricci curvature bounds

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves Willmore-type inequalities for bounded domains in complete non-compact manifolds under asymptotic or integral Ricci curvature bounds, continuously recovering the pointwise Ricci lower-bound case.

desk verdict A genuinely new pair of Willmore-type inequalities under asymptotic and L^p integral Ricci bounds, built on sound comparison estimates; the main theorem in Section 3 has two repairable slips in the write-up, not in the underlying argument. read the letter →

arxiv 2508.20298 v1 pith:WXFOMTHN submitted 2025-08-27 math.DG

Jihye Lee This is my paper · ORCID

classification math.DG MSC 53A0753C21

keywords WillmoreinequalityintegralRiccicurvaturemeanrelativevolumeratiocomparisongeometryhypersurfaceslowerboundtubularneighborhoods

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 establishes that Willmore-type inequalities—lower bounds on the total bending of a boundary in terms of its mean curvature—hold on complete non-compact manifolds whose Ricci curvature is controlled only in an averaged or asymptotically decaying sense. The main theorems bound the relative volume ratio of a bounded domain by an integral over its boundary of $(1+H/n)^n$, with an explicit error term that vanishes as the integral Ricci curvature defect goes to zero. When the defect is identically zero, both inequalities recover the recent Jin–Yin theorem under the pointwise bound $\mathrm{Ric}\ge -n$. The interest is that the proof works without any pointwise lower curvature bound, relying instead on an $L^p$ norm of the negative part of Ricci for $p>(n+1)/2$.

What carries the argument

The engine is a Riccati comparison for the mean curvature $m(t)$ of parallel hypersurfaces along normal geodesics, $m'+m^2/n\le n+\rho(\gamma(t))$. In the asymptotic case, two comparison lemmas for the linear ODE $\psi''=(1+\Lambda(t))\psi$ give sharp control on the ratios $\psi_2/\psi_1$ and $\psi_1/\sinh t$, yielding a pointwise Jacobian bound of the form $(\cosh t+(2b+H/n)\sinh t)^n e^{2nb}$. In the integral case, the deviation $\phi=\max\{m-\hat m,0\}$ from the hyperbolic comparison mean curvature is inserted into an $L^p$ estimate: a Hölder argument bounds $\int_0^r \phi^{2p}J\,dt$ by a constant times $\int_0^r \rho^p J\,dt$, and a second Hölder step converts this into a multiplicative correction to the hyperbolic Jacobian $\hat J$. Lemma 3.1, the elementary inequality $(1+b)^p\le 1+\varepsilon+C(p,q,\varepsilon)\varepsilon^{-q}b^p$ with $C(p,q,\varepsilon)\to0$ as $\varepsilon\to0$, controls the error terms; choosing $\varepsilon=\|\rho\|_p^{1/2}$ produces the final bound.

What would settle it

Construct an explicit family of complete non-compact metrics $g_\varepsilon$ on a fixed manifold with $\|\rho\|_p(g_\varepsilon)\to0$ and a fixed bounded mean-convex domain $\Omega$ whose Jacobi fields are computable—for instance a warped product $M=\mathbb{R}\times_\varphi \Sigma$ with $\rho$ explicit—and compute both sides of Theorem 1.3. If the left side minus the boundary integral exceeds the sum of the $\varepsilon^{1/2}$-term and the vanishing $C$-term for some $\varepsilon$, the theorem fails; if the excess is always bounded by the predicted rate, it supports sharpness of the choice $\varepsilon=\|\rho\|_p^{1/2}$.

Watch

Extended reading notes

Core claim

The central discovery is that a Willmore-type inequality of the form $$\mathrm{RV}(\$\Omega$)\cdot\omega_n \le \left(1+\|\rho\|$_p^{{1/2}}$\right)\int_{\partial\$\Omega$}\left(1+\frac{H}{n}\right)^n d\mathrm{vol} + C(n,p,\|\rho\|_p)\left(1+\frac{H(\xi)}{n}\right)^n$$ holds under the sole assumption that $\rho=\max\{-n-\mathrm{Ric},0\}$ lies in $L^p$ with $p>(n+1)/2$, provided the domain is mean-convex and has finite relative volume ratio, and with $C(n,p,\|\rho\|_p)\to 0$ as $\|\rho\|_p\to 0$. In the asymptotic setting, a pointwise but decaying bound $\mathrm{Ric}\ge -n-n\lambda(d(o,\cdot))$ with $\lambda\in L^1$ non-increasing yields a similar inequality with the explicit factor $e^{2nb}$ and integration over the region where $H\ge -n-2nb$. Both results recover the Jin–Yin theorem when the curvature defect is identically zero.

Load-bearing premise

The load-bearing premise is mean-convexity of the boundary, $H(x)\ge0$ on $\partial\Omega$, together with the separate assumption that $\mathrm{RV}(\Omega)$ is finite, since both are needed for the key estimates and neither is derived under the integral curvature hypothesis.

Editorial extensions

If this is right

When $\rho\equiv 0$, Theorem 1.3 reduces exactly to the Jin–Yin Willmore-type inequality for $\mathrm{Ric}\ge -n$, with no error term.
The inequality is quantitatively stable: as $\|\rho\|_p\to0$, the correction term disappears and the bound converges continuously to the pointwise one.
The asymptotic Theorem 1.1 implies that $\partial\Omega\cap\{H\ge -n-2nb\}$ is non-empty for every bounded domain, and it yields a compactness criterion for manifolds with boundary whose mean curvature is bounded below.
Under the asymptotic assumptions, the finiteness of $\mathrm{RV}(\Omega)$ follows from the curvature bound, so Theorem 1.1 needs no separate finiteness hypothesis.
The integral setting requires mean-convexity of the boundary; without $H\ge0$ on $\partial\Omega$, the key non-negativity step in the proof fails.

Reading between the lines

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

The same Riccati-comparison strategy should extend to intermediate Ricci curvature bounds, replacing the ambient dimension $n+1$ by $k+1$ and adjusting the critical exponent accordingly, since only control along normal geodesics is used.
The specific choice $\varepsilon=\|\rho\|_p^{1/2}$ in the proof suggests the error term is at most of order $\|\rho\|_p^{1/2}$; explicit warped-product examples could test whether this exponent is sharp.
A proof that $\mathrm{RV}(\Omega)<\infty$ follows directly from an integral Ricci bound alone would remove one of the two technical assumptions in Theorem 1.3 and would likely require a volume comparison theorem for tubular neighborhoods under integral curvature.
The theorem quantifies how much the Willmore constant can deteriorate per unit of averaged Ricci curvature below $-n$, which is a natural input for convergence or collapse questions in Riemannian geometry.

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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

The two main theorems are new and probably correct. Theorem 1.1 handles asymptotic Ricci decay with λ∈L^1, which is weaker than the usual ∫tλ<∞ condition, and Theorem 1.3 gets a Willmore inequality from ∥ρ∥_p with an error term that vanishes as ∥ρ∥_p→0. Both recover Jin–Yin. That is a real step beyond Gallot and Petersen–Sprouse, and the ODE lemmas in Section 2 are clean and well proved.

The soft spots are in the write-up, mostly in Section 3. The line before (3.3) has a wrong exponent: the author multiplies (J/bJ)' ≤ φ(J/bJ) by (J/bJ)^{1/(2p−1)}, but the integrated inequality that follows needs the multiplier (J/bJ)^{1/(2p)−1}. The result is fine, but as displayed it is algebraically inconsistent. The second issue is that the constant C(n,p) defined after (3.3) depends on x through H(x) in bJ. It is then pulled outside the ∂Ω integral without comment. Under H≥0 it is uniformly bounded by C(n,p,0), so the gap is easy to close, but a referee will want it stated. The reader's minor point about Theorem 2.6 is also real: the dominating function used in dominated convergence can be negative when H<−n−2nb; use the positive part or split the boundary into the two regions. None of this undermines the main claims, in my view.

Citation pattern is honest: the work builds on [12] and [16], and the two self-citations are not load-bearing. The paper is not circular and does not hide anything. Presentation needs a careful proofread—there are typos and the constant-dependence issue obscures the proof.

This paper is for people working in comparison geometry and Willmore-type inequalities. My recommendation: send it to review. The theorems are worth referee time, and the repairs are local. I'd ask the author to fix the exponent typo, justify the uniform bound on C(n,p,H), and patch the dominated convergence argument. After that, I'd expect it to be acceptable.

Referee Report

2 major / 4 minor

Summary. The paper proves two Willmore-type inequalities for bounded domains with smooth boundary in complete non-compact (n+1)-manifolds. Theorem 1.1 assumes an asymptotic Ricci lower bound Ric ≥ −n − λ(d(o,·)) with non-increasing λ ∈ L^1, and bounds RV(Ω)ω_n by an integral over {H ≥ −n−2nb} of (1+2b+H/n)^n times e^{2nb}. Theorem 1.3 assumes an L^p integral Ricci bound with p > (n+1)/2, mean-convex boundary, and finite RV(Ω), and bounds RV(Ω)ω_n by (1 + ||ρ||_p^{1/2})∫_{∂Ω}(1+H/n)^n plus an error term that tends to 0 as ||ρ||_p → 0. The proofs use ODE comparison lemmas for ψ_1, ψ_2, a Riccati comparison for the mean curvature of parallel hypersurfaces, Jacobian estimates, and Hölder estimates with a boundedness lemma (Lemma 3.1). The paper also contains corollaries on non-emptiness of {H ≥ −n−2nb} and a compactness criterion for manifolds with boundary.

Significance. If the proof of Theorem 1.3 is corrected, the main results are a genuine extension of the Jin-Yin Willmore inequality to integral and asymptotic Ricci curvature settings, with a quantitative error term in the integral case. The ODE comparison arguments in Section 2 are elementary, clearly presented, and appear correct; they also yield finiteness of the relative volume ratio under asymptotic assumptions. The paper is self-contained modulo standard comparison theorems and does not rely on circular reasoning. The main weakness is that the key Jacobian estimate in Section 3 is not derived correctly as written, and one constant depends on the point x; both issues are locally repairable, so the central claim remains plausible.

major comments (2)

[Section 3, display preceding (3.3)] The exponent manipulation that yields the integrated Jacobian estimate is algebraically inconsistent. From Z := J/bJ satisfying Z' ≤ φZ, multiplying by Z^{1/(2p−1)} gives Z^{1/(2p−1)}Z' ≤ φ Z^{2p/(2p−1)}, not φ Z^{1/(2p)}. The integrated inequality 2p(Z^{1/(2p)}−1) ≤ ∫ φ Z^{1/(2p)} dt is exactly what follows from (d/dt)Z^{1/(2p)} ≤ (1/(2p))φ Z^{1/(2p)}, whose left factor is Z^{1/(2p)−1}. The displayed line should be corrected; the intended bound is nevertheless obtainable with the correct exponent, so this is repairable.
[Section 3, definition of C(n,p) after (3.3)] The constant C(n,p) depends on x because bJ(x,t) contains H(x), yet it is pulled out of the Σ-integral in the subsequent volume estimate. Since H ≥ 0, one can bound bJ(x,t)^{-1/(2p−1)} ≤ (cosh t)^{-n/(2p−1)}, giving a uniform constant independent of x; this step needs to be stated explicitly. As written, the pull-out is unjustified and must be repaired for the proof of Theorem 1.3 to be complete.

minor comments (4)

[Theorem 2.6 proof] The quantity \bar J is used in the integrand without being defined; it should be defined explicitly, presumably as the Jacobian determinant J or its normalized extension.
[Theorem 2.6 proof] The dominance bound (coth t_0 + 2b + H(x)/n)^n can be negative when H(x) is very negative. Since H is bounded on the compact boundary, the proof should choose t_0 after bounding H below, or replace the displayed quantity by its positive part, so that the dominated convergence argument is valid.
[Corollary 2.4] There is a typo: 'Riemaninan' should be 'Riemannian'.
[Lemma 3.1 proof] The proof of Lemma 3.1 could state more explicitly that F_{p,q,ε} attains its maximum on (0,∞) because the limits at 0 and ∞ are finite; the current wording is terse but correct.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and uses external benchmarks.

full rationale

The paper's central claims are obtained from a Riccati comparison argument, elementary ODE lemmas, and standard comparison theorems, with no parameter fitted to the target inequalities. In Theorem 1.1, the estimates (2.12), (2.13), and (2.17) are derived from Lemmas 2.1-2.3, whose proofs use the ODE comparison theorem from [18] and the comparison lemma quoted from [8]; the relative volume ratio is not defined in terms of the Willmore functional. In Theorem 1.3, the error constant C(n,p,||rho||_p) is constructed from Lemma 3.1 and Hölder estimates, and it is not a fitted parameter; the endpoint case ||rho||_p=0 is handled by the independent external theorem [12, Theorem 1.3], not by the author's own prior work. The only self-citations ([14], [15]) appear in a survey remark about the p>n/2 threshold alongside independent references [6,7,16], and they are not load-bearing. No 'prediction' in the paper reduces by construction to its input: the mean-convexity and RV(Omega)<infinity assumptions restrict the setup but are not tailored to force the conclusion. The algebraic inconsistency in the displayed Jacobian estimate noted by a skeptical reader concerns correctness of a repairable step, not circularity, and does not change the circularity assessment.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical parameters are fitted to data. All constants in the inequalities depend only on n, p, q, epsilon, b, or ||rho||_p and are either structural or auxiliary variables introduced in the estimates; none is tuned to make the conclusion hold by construction. No new geometric objects, particles, or forces are introduced; the comparison functions bm and bJ are standard hyperbolic-space quantities, and rho is the standard negative part of the Ricci curvature.

assumptions (5)

standard math Second-order linear ODE comparison theorem (Pigola-Rigoli-Setti, Lemma 2.1) applied to solutions psi1, psi2 and supersolution phi2.
Used in Lemma 2.1 and the proof of Theorem 1.1 to convert the Riccati inequality into comparison of psi and phi.
standard math Jacobi field and Riccati equation for mean curvature of parallel hypersurfaces, including m'(t)+trS+(1/n)m^2 <= -Ric(gamma',gamma').
Central equation (2.10) and (3.1); standard in comparison geometry.
standard math Cut locus decomposition and area formula for the normal exponential map on a smooth boundary.
Used to express volume of tubular neighborhoods as an integral of the determinant of Jacobi fields in the proof of Theorem 1.1.
standard math Dominated convergence and L'Hopital's rule in Theorem 2.6.
Used to pass from limits of integrands to the relative volume ratio limit.
domain assumption Curvature hypotheses: Ric>=-n-lambda(d(o,.)) with lambda nonincreasing in L^1 for Theorem 1.1; rho in L^p(M) with p>(n+1)/2 and RV(Omega)<infinity for Theorem 1.3.
These are the paper's stated hypotheses; monotonicity of lambda is used to bound integral_0^infinity lambda(d(o,gamma(t)))dt <= 2b.

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Pith. "Pith review of A Willmore-type inequality for hypersurfaces with asymptotic or integral Ricci curvature bounds." pith.science (2026). https://pith.science/paper/WXFOMTHN

@misc{pith2026250820298,
  author       = {Pith},
  title        = {Pith review of: A Willmore-type inequality for hypersurfaces with asymptotic or integral Ricci curvature bounds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WXFOMTHN}},
  note         = {Machine review of arXiv:2508.20298}
}

read the original abstract

We establish Willmore-type inequalities for bounded domains in complete non-compact Riemannian manifolds, under either asymptotic or integral Ricci curvature bounds. Those results recover a recent inequality of Jin-Yin arXiv:2402.02465.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [12]

Willmore-type inequality for closed hypersur- faces in complete manifolds with Ricci curvature bounded below

Xiaoshang Jin and Jiabin Yin. Willmore-type inequality for closed hypersur- faces in complete manifolds with Ricci curvature bounded below. arXiv preprint arXiv:2402.02465, 2024

arXiv 2024

[2] [16]

Petersen and G

P. Petersen and G. Wei. Relative volume comparison with integral curvature bounds. Geom. Funct. Anal., 7(6):1031–1045, 1997. 20

work page 1997

[3] [1]

Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

Virginia Agostiniani, Mattia Fogagnolo, and Lorenzo Mazzieri. Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature. Invent. Math., 222(3):1033–1101, 2020

work page 2020

[4] [2]

Finiteness of π1 and geometric inequalities in almost positive Ricci curvature

Erwann Aubry. Finiteness of π1 and geometric inequalities in almost positive Ricci curvature. Ann. Sci. ´Ecole Norm. Sup. (4) , 40(4):675–695, 2007. 19

work page 2007

[5] [3]

Comparison geometry for substatic man- ifolds and a weighted isoperimetric inequality

Stefano Borghini and Mattia Fogagnolo. Comparison geometry for substatic man- ifolds and a weighted isoperimetric inequality. arXiv preprint arXiv:2307.14618 , 2023

arXiv 2023

[6] [4]

Galloway

Mingliang Cai and Gregory J. Galloway. Boundaries of zero scalar curvature in the AdS/CFT correspondence. Adv. Theor. Math. Phys. , 3(6):1769–1783, 1999

work page 1999

[7] [5]

On a theorem of Fenchel-Borsuk-Willmore-Chern-Lashof

Bang-yen Chen. On a theorem of Fenchel-Borsuk-Willmore-Chern-Lashof. Math. Ann., 194:19–26, 1971

work page 1971

[8] [6]

Improved relative volume comparison for integral Ricci curvature and applications to volume entropy

Lina Chen and Guofang Wei. Improved relative volume comparison for integral Ricci curvature and applications to volume entropy. J. Geom. Anal., 32(1):Paper No. 2, 17, 2022

work page 2022