Fenchel-Willmore-Chen inequality under lower bounds on weighted intermediate Ricci curvature

Guofang Wei; Jihye Lee

REVIEW 3 major objections 5 minor 22 references

Fenchel-Willmore-Chen inequality under lower bounds on weighted intermediate Ricci curvature

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves a sharp scale-invariant Fenchel–Willmore–Chen inequality for closed submanifolds in complete weighted Riemannian manifolds with a lower bound on weighted intermediate Ricci curvature; the lower bound is the weighted…

desk verdict A valuable unified Fenchel–Willmore–Chen inequality whose proof is conditional on an unproved estimate imported from an unreviewed preprint; worth refereeing. read the letter →

arxiv 2608.01171 v1 pith:LW3H5SAB submitted 2026-08-02 math.DG

Jihye Lee , Guofang Wei This is my paper · ORCID

classification math.DG MSC 53C2153C4253C24

keywords Fenchel–Willmore–CheninequalitysmoothmetricmeasurespaceweightedRiccicurvatureBakry–EmeryintermediatemeanisoperimetricSobolev

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 a sharp Fenchel–Willmore–Chen inequality for closed submanifolds in smooth metric measure spaces (weighted Riemannian manifolds) under a lower bound on the 1-Bakry–Emery k-Ricci curvature. The bound expresses the integral of the n-th power of a weighted mean curvature vector, corrected by the gradient of the weight, in terms of a weighted relative volume ratio of the submanifold. The result extends the classical Euclidean inequality and unifies a number of recent extensions to nonnegative Ricci, intermediate Ricci, and substatic settings, while also covering negative curvature bounds. The proof compares the Jacobian of the normal exponential map with a model quantity defined through a weight-reparameterized distance, and it yields an equality case with a warped-product rigidity. A Sobolev inequality and an isoperimetric inequality for nonnegative 1-weighted Ricci curvature follow as corollaries.

What carries the argument

The proof is carried by three mechanisms. The first is the weight-reparameterized distance $s(t)=\int_0^t e^{-2f(\gamma(\tau))/(n+m-1)} d\tau$, which converts the curvature assumption $\mathrm{Ric}^1_{n,f}\ge nKe^{-4f/(n+m-1)}$ into the standard Riccati and Jacobi comparisons in the $s$-variable; in this variable the model volume element is $\mathrm{sn}_K^{n+m-1}(s)\,ds$, so the weight is absorbed into the geometry. The second is the factorization of the Jacobian of the normal exponential map, $|\det D\Phi(x,y,t)| = t^{m-1}|\det(D\exp_x)_{ty}|\,|\det Q_t|$, with $Q_t = \frac12 \mathrm{Hess}\, d^2_{\gamma(t)}(x)|_{T_x\Sigma\times T_x\Sigma} - \langle \mathrm{II}_x, ty\rangle$; the determinant $|\det Q_t|$ is bounded above through the partial Hessian comparison and an arithmetic-geometric-mean step. The third is a pair of monotone quantities, $\theta_1$ and $\theta_2$, whose monotonicity yields the sharp pointwise Jacobian bound (1.14); these are integrated over the $s$-tubular neighborhood and passed to the limit $S\to\infty$ to produce the weighted relative volume ratio on the right-hand side.

What would settle it

Compute both sides of Theorem 0.1 in an explicit weighted model space — for instance hyperbolic space with weight $f=c\,d(\cdot,p)$ and $\Sigma$ a small round sphere — and check whether the claimed integral strictly bounds $\mathrm{RV}_{\mu,K}(\Sigma)$; any configuration with the left side smaller than the right side would falsify the theorem. Alternatively, test the quoted differential inequality (1.10) directly by computing $|\det Q_t|$ and the Jacobi-field product for a short normal geodesic in a rank-one symmetric space with nonconstant $f$; a counterexample to (1.10) would falsify the presented proof.

Watch

Extended reading notes

Core claim

At the heart of the paper is Theorem 0.1: if $(M^{n+m}, g, e^{-f}d\mathrm{vol})$ is a complete noncompact smooth metric measure space whose weighted intermediate Ricci curvature obeys $\mathrm{Ric}^1_{n,f}(v,P) \ge n K e^{-4f/(n+m-1)}$ with $K \le 0$, then every closed immersed $n$-dimensional submanifold $\Sigma$ satisfies $\int_\Sigma e^{f(x)} \int_{S^\perp_x \Sigma} (\sqrt{-K} e^{-2f(x)/(n+m-1)} + \langle -\vec H_f(x), y\rangle)_+^n dy \, d\sigma(x) \ge \mathrm{RV}_{\mu,K}(\Sigma)$, where $\vec H_f = \vec H + (\nabla f)^\perp/(n+m-1)$ is the weighted mean curvature vector and $\mathrm{RV}_{\mu,K}(\Sigma)$ is the weighted relative volume ratio built from the $s$-tubular volume of $\Sigma$. The inequality is sharp and scale-invariant: in the Euclidean, unweighted case it reduces to $\int_\Sigma |\vec H|^n d\sigma \ge \mathrm{vol}(S^n)$, and the equality case is rigid, with the pullback of the ambient metric under the normal exponential map taking the warped-product form $(a_t b_t)^2 g_H + a_t^2 g_V + dt^2$. The curvature assumption involves the dimension of $\Sigma$ and not the codimension, a feature inherited from the Jacobian decomposition used in the proof.

Load-bearing premise

The load-bearing premise is an unproved differential inequality for the Jacobian determinant of the normal exponential map, quoted from a companion paper; if that inequality or the associated factorization lemma does not hold for the weighted normal exponential map with the stated coefficients, the central inequality does not follow from the presented argument.

Editorial extensions

If this is right

In the unweighted case $f=0$ and $K=0$, Theorem 0.1 recovers the classical sharp bound $\int_\Sigma |\vec H|^n d\sigma \ge \mathrm{vol}(S^n)$, with equality constraining $\Sigma$ to be an umbilical hypersphere in the Euclidean case.
For $K<0$, the inequality remains explicit and non-vacuous: the weight $\sqrt{-K}\,e^{-2f(x)/(n+m-1)}$ appears in the integrand, giving a new bound even in negatively curved weighted ambient spaces.
The equality case (Proposition 1.10) implies that a submanifold saturating the bound has an infinite normal cut time in the relevant directions and that the normal exponential map pulls back the metric to a warped product — a strong rigidity statement for submanifolds achieving the volume ratio.
The Sobolev inequality (Theorem 2.1) with $\mathrm{Ric}^1_f \ge 0$ and $f$ bounded below implies the weighted isoperimetric inequality $\mathrm{vol}_f(\partial\Omega) \ge n \theta_f^{1/n} e^{2\inf f/n} (\mu_f(\Omega))^{(n-1)/n}$, extending the nonnegative-curvature isoperimetric result to the weighted setting.

Reading between the lines

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

Because the curvature condition and the $s$-parameter involve no reference to the codimension, the same Jacobian-factorization strategy could plausibly handle $\mathrm{Ric}^1_{\ell,f}$ with $\ell$ between 1 and $n$, producing a family of intermediate sharp inequalities; this is not pursued in the paper.
The proof quotes the key differential inequality (1.10) from a companion paper; a direct derivation of that inequality from the weighted Jacobi equation would make the argument self-contained and could clarify the role of the factor $n$.
A numerical check of the inequality in a model weighted space (for instance hyperbolic space with a radial weight and a round equatorial sphere) could test whether equality occurs outside the rigid warped-product configuration; if it does, Proposition 1.10 would require modification.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Dear colleague,

The main theorem is a genuine consolidation. It subsumes the recent Fenchel-Willmore-Chen inequalities for nonnegative k-Ricci curvature (Ji-Kwong, Pan-Yi), weighted Ric-infinity (Wu-Wu), and the substatic case (Borghini-Fogagnolo) under one lower bound on the 1-Bakry-Emery n-Ricci curvature, and it gives the correct power in the volume ratio. The proof strategy is standard comparison geometry, executed carefully; the reparameterized distance is a clean trick, and the equality/rigidity analysis in Section 1 goes well beyond what is usual in this literature. The Sobolev/isoperimetric half is a solid, if modest, improvement over Fujitani-Sakurai.

Three soft spots, in decreasing order of severity.

1. The determinant estimate for Q_t — the technical heart of the paper — is gated by inequality (1.10), quoted from the unreviewed preprint [14, Eq. (3.24)], plus the factorization [14, Lemma 3.2] and the positive-definiteness [14, Lemma 3.4(i)]. None of these is proved in the present paper. If any of them carries a hidden hypothesis, the Jacobian bound (1.12) and hence Theorem 0.1 would not follow. This is not circularity; it is an unverified dependency. The referee should require a complete proof of (1.10), or at the very least a precise statement of the boundary conditions and semi-positivity assumptions under which it was derived.

2. The notation conflates the submanifold dimension n with the ambient dimension n+m. Proposition 1.1 says M^n, then Proposition 1.3 applies it to M^{n+m} with ℓ=n; denominators and curvature coefficients shift accordingly. The math is consistent if one reads the proposition's n as the ambient dimension, but as written it is hard to follow. Fixable, but it needs to be rewritten.

3. In Section 2, the step from the limsup estimate to ∫ φ^{n/(n-1)} ≥ θ_f e^{2 inf f} is asserted without proof. There is an implicit asymptotic identification of the volume of the 'everywhere within distance r' set with θ_f; the algebra around the AM-GM step also needs a careful check. This is probably repairable, but as written it is a gap.

My overall assessment: the central claim is likely correct, and if (1.10) can be verified, this is an important paper. The flaws are specific and addressable. Send it to a serious referee before accepting.

Best

Referee Report

3 major / 5 minor

Summary. The paper proves a Fenchel–Willmore–Chen type inequality for closed immersed submanifolds of a complete noncompact smooth metric measure space with a lower bound on the 1-Bakry–Emery n-Ricci curvature, and derives corollaries recovering Chen's classical inequality, the zero-curvature weighted case, and a hypersurface version for K=-1. The proof uses the normal exponential map, a factorization of its Jacobian from an external preprint, a weighted partial Hessian comparison via a reparametrized distance, and Jacobi-field determinant estimates. A second part proves a Sobolev inequality and an isoperimetric inequality under the assumption Ric^1_f ≥ 0.

Significance. If correct, the main theorem would be a substantial unification and extension of known Fenchel–Willmore–Chen inequalities to weighted manifolds with intermediate Ricci curvature bounds, and the paper is honestly and explicitly connected to prior work. The use of the reparametrized distance to absorb the weight function is natural, the Riccati-comparison structure is standard, and the corollaries are meaningful. The main theorem, however, rests on at least one externally imported inequality whose hypotheses are not stated or proved, and there is a systematic notational dimension slip that must be repaired before the proof can be accepted. The Sobolev/isoperimetric part is also of interest if the comparison argument is completed.

major comments (3)

[Section 1.2, Propositions 1.1 and 1.5; application in Propositions 1.3 and 1.7] Propositions 1.1 and 1.5 are stated for an n-dimensional ambient manifold (M^n,g,e^{-f}dvol) with ℓ ≤ n-1, but Theorem 0.1 is set in an (n+m)-dimensional ambient manifold and applies these propositions with ℓ=n to the submanifold tangent plane. In the proof of Proposition 1.3, Proposition 1.1 is invoked with ℓ=n and a denominator n+m-1; in the paragraph following Proposition 1.5, Proposition 1.5 is invoked with ℓ=n and the same denominator. As written, these applications are outside the stated hypotheses because the propositions use n-1 in the exponent and in the coefficient of f'. The correct statements are obtained by re-stating the propositions for an ambient dimension N with ℓ≤N-1 and denominators N-1, which is what the later applications require. Please correct this conflation; I do not see a substantive failure once the propositions are re-stated for a general ambient dimension.
[Section 1.3, inequality (1.10)] The load-bearing determinant estimate (1.10) is quoted verbatim from the unreviewed preprint [14, Eq. (3.24)] without proof. The authors do not state the hypotheses under which (1.10) is derived in [14], such as the boundary conditions on the Jacobi fields J_i^{t_0}, the required positive-definiteness of A_t, and whether any curvature lower bound enters. The monotonicity of θ_1 in (1.11) and the Jacobian estimate (1.12) depend directly on (1.10), so Theorem 0.1 is conditional on an unverified external result. Please either include a full proof of (1.10) in this paper or state and prove the precise hypotheses and give a self-contained derivation.
[Section 1.3, proof of Proposition 1.3] The proof of Proposition 1.3 applies the arithmetic–geometric mean inequality to the eigenvalues of A_{t_0}, which requires A_{t_0} to be positive definite. Positive definiteness is imported from [14, Lemma 3.4(i)] without proof. Because the determinant estimate (1.12) and the subsequent volume comparison collapse if A_{t_0} fails to be positive definite for some t_0<T_cut, this input is load-bearing and should be established within the paper or replaced by a direct argument.

minor comments (5)

[Abstract and title header] The title header contains broken spacing: 'INTERMEDIA TE RICCI CUR V A TURE' should be 'INTERMEDIATE RICCI CURVATURE'.
[Corollary 0.5] The phrase 'smooth metric measurement space' should be 'smooth metric measure space'.
[Section 1.5, Lemma 1.9] The proof of Lemma 1.9 invokes continuity of the normal cut-time function without a reference or proof; please add a justification or a citation.
[Equations (1.16) and (1.17)] The notation for the measure on S_x^⊥Σ is not uniform: dω_{S_x^⊥Σ}(y) in the tube formula becomes simply dy in (1.16)–(1.17). Please use one notation consistently.
[Section 1.4, equation after (1.13)] The limit for the volume-density comparison is written as t→0^+, but the monotonicity of θ_2 is established in the reparametrized variable s; please spell out the change of variables in the limit computation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the theorem is a genuine comparison inequality, and all load-bearing imports are external non-self citations.

full rationale

The derivation chain starts from the curvature assumption Ric^1_{n,f}(v,P) >= n K e^{-4f/(n+m-1)} and uses the normal exponential map, the factorization |detDExp^⊥| = |det(Dexp_x)_v||det Q_v| from [14], the weighted partial Hessian comparison (Proposition 1.1), and the imported differential inequality (1.10) to prove monotonicity of θ_1 and the Jacobian bound (1.12). None of these steps is definitionally equivalent to the conclusion: θ_1 and θ_2 are monotone quotients whose initial limits are fixed asymptotics, not free parameters fitted to the target integral. The right-hand side RV_{μ,K}(Σ) is independently defined in (1.16) as a limsup of μ_f-volumes of s-tubes divided by the model volume ∫ sn_K^{n+m-1}; it is not constructed from the left-hand integral. The proof then integrates (1.14), divides by the model volume, and takes a limit, which genuinely yields the claimed inequality. The main external inputs [14] and [22] are preprints/citations by other authors, not self-citations, so the self-citation patterns do not apply; the only self-citation [13] is contextual and not load-bearing. Whether [14, Eq. (3.24)] is correct is a correctness risk, not a circularity, because the paper does not redefine its target as that inequality or fit parameters to force it.

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

The paper introduces no free parameters or new physical entities. Its central claim rests on two external geometric lemmas from [14] and [7], plus standard elliptic theory and on an asymptotic identification in Section 2 that is asserted without proof.

assumptions (4)

domain assumption Pan-Yi's normal exponential Jacobian factorization and differential inequality (1.10)
Quoted from [14, Lemma 3.2 and Eq (3.24)] without proof. This is load-bearing for Proposition 1.3 and the monotonicity of θ_1 in (1.11), which drives the main theorem.
domain assumption Inner parallel set volume ratio tends to θ_f in the Sobolev proof
Used in Section 2 to conclude ∫ φ^{n/(n-1)} dvol_f ≥ θ_f e^{2inf f}. This follows from Bishop-Gromov for compact Ω, but the paper does not state or prove it.
standard math Existence and C^{2,α} regularity for the weighted Neumann problem in Theorem 2.1
Invoked from standard elliptic theory to obtain the solution u used in the ABP argument.
domain assumption Containment lemma of Chong-Luo-Lu [7, Lemma 2.2] (Lemma 2.3 here)
Imported from [7] to relate the ABP map Φ_r(A_r) to the inner parallel set. The paper restates it without proof.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Fenchel-Willmore-Chen inequality under lower bounds on weighted intermediate Ricci curvature." pith.science (2026). https://pith.science/paper/LW3H5SAB

@misc{pith2026260801171,
  author       = {Pith},
  title        = {Pith review of: Fenchel-Willmore-Chen inequality under lower bounds on weighted intermediate Ricci curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LW3H5SAB}},
  note         = {Machine review of arXiv:2608.01171}
}

read the original abstract

We establish a Fenchel-Willmore-Chen inequality for smooth metric measure spaces with a lower bound on the 1-Bakry-Emery k-Ricci curvature. This extends previous results and yields a unified proof. In addition, a Sobolev and isoperimetric inequality for nonnegative 1-weighted Ricci curvature is proven.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 19 canonical work pages

[14]

A comparison theorem with applications to sharp geometric inequalities for submanifolds

Shengliang Pan and Chengyang Yi. A comparison theorem with applications to sharp geo- metric inequalities for submanifolds.arXiv:2605.06074v2, 2026

work page Pith review arXiv 2026
[1]

Sharp geometric inequali- ties for closed hypersurfaces in manifolds with nonnegative Ricci curvature.Invent

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

work page 2020
[2]

Comparison geometry for substatic manifolds and a weighted isoperimetric inequality.arXiv:2307.14618, 2023

Stefano Borghini and Mattia Fogagnolo. Comparison geometry for substatic manifolds and a weighted isoperimetric inequality.arXiv:2307.14618, 2023

arXiv 2023
[3]

S. Brendle. Geometric inequalities and the Alexandrov-Bakelman-Pucci technique.Science China Mathematics, https://doi.org/10.1007/s11425-026-2652-7, 2026

work page doi:10.1007/s11425-026-2652-7 2026
[4]

Sobolev inequalities in manifolds with nonnegative curvature.Comm

Simon Brendle. Sobolev inequalities in manifolds with nonnegative curvature.Comm. Pure Appl. Math., 76(9):2192–2218, 2023

work page 2023
[5]

Yousef K. Chahine. Volume estimates for tubes around submanifolds using integral curvature bounds.J. Geom. Anal., 30(4):4071–4091, 2020

work page 2020
[6]

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

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

work page 1971
[7]

Sobolev Inequality In Manifolds With Lower Quadratic Curvature Decay

Tian Chong, Han Luo, and Lingen Lu. Sobolev inequality in manifolds with lower quadratic curvature decay.arXiv preprint arXiv:2503.19254, 2025

work page Pith review arXiv 2025

Show all 22 references

[8]

¨Uber Kr¨ ummung und Windung geschlossener Raumkurven.Math

Werner Fenchel. ¨Uber Kr¨ ummung und Windung geschlossener Raumkurven.Math. Ann., 101(1):238–252, 1929

1929
[9]

Geometric analysis on weighted manifolds under lower 0-weighted Ricci curvature bounds.Nonlinear Anal., 263:Paper No

Yasuaki Fujitani and Yohei Sakurai. Geometric analysis on weighted manifolds under lower 0-weighted Ricci curvature bounds.Nonlinear Anal., 263:Paper No. 113965, 15, 2026. 23

2026
[10]

A general comparison theorem with applications to volume estimates for submanifolds.Ann

Ernst Heintze and Hermann Karcher. A general comparison theorem with applications to volume estimates for submanifolds.Ann. Sci. ´Ecole Norm. Sup. (4), 11(4):451–470, 1978

1978
[11]

Fenchel-Willmore inequality for submanifolds in manifolds with non-negativek-Ricci curvature.arXiv:2507.07655, 2025

Meng Ji and Kwok-Kun Kwong. Fenchel-Willmore inequality for submanifolds in manifolds with non-negativek-Ricci curvature.arXiv:2507.07655, 2025

2025 arXiv
[12]

Willmore-type inequality for closed hypersurfaces in complete manifolds with Ricci curvature bounded below.arXiv:2402.02465, 2024

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

2024 arXiv
[13]

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

Jihye Lee. A Willmore-type inequality for hypersurfaces with asymptotic or integral Ricci curvature bounds.Ann. Global Anal. Geom., 70(2):6, 2026

2026
[15]

Springer, 2016

Peter Petersen.Riemannian geometry. Springer, 2016

2016
[16]

A sharp geometric inequality for closed hypersurfaces in manifolds with asymptotically nonnegative curvature.arXiv:2310.08245, 2023

Adam Rudnik. A sharp geometric inequality for closed hypersurfaces in manifolds with asymptotically nonnegative curvature.arXiv:2310.08245, 2023

2023 arXiv
[17]

Remark on a geometric inequality for closed hypersurfaces in weighted mani- folds.arXiv:2510.26062, 2025

Adam Rudnik. Remark on a geometric inequality for closed hypersurfaces in weighted mani- folds.arXiv:2510.26062, 2025

2025
[18]

Remark on an inequality for closed hypersurfaces in complete manifolds with nonnegative Ricci curvature.Ann

Xiaodong Wang. Remark on an inequality for closed hypersurfaces in complete manifolds with nonnegative Ricci curvature.Ann. Fac. Sci. Toulouse Math. (6), 32(1):173–178, 2023

2023
[19]

Al. I. Cuza

T. J. Willmore. Mean curvature of immersed surfaces.An. S ¸ti. Univ. “Al. I. Cuza” Ia¸ si Sect ¸. I a Mat. (N.S.), 14:99–103, 1968

1968
[20]

Willmore-type inequalities for closed hypersurfaces in weighted manifolds.J

Guoqiang Wu and Jia-Yong Wu. Willmore-type inequalities for closed hypersurfaces in weighted manifolds.J. Geom. Anal., 35(11):Paper No. 345, 40, 2025

2025
[21]

Sectional curvature for Riemannian manifolds with density.Geom

William Wylie. Sectional curvature for Riemannian manifolds with density.Geom. Dedicata, 178:151–169, 2015

2015
[22]

On the geometry of riemannian manifolds with density

William Wylie and Dmytro Yeroshkin. On the geometry of riemannian manifolds with density. arXiv:1602.08000, 2016. (Jihye Lee)School of Mathematical and Physical Sciences, Macquarie University, Sydney, NSW, Australia Email address:jihye.lee@mq.edu.au (Guofang Wei)Department of ...

2016 arXiv

Pith tools

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

[1] [14]

A comparison theorem with applications to sharp geometric inequalities for submanifolds

Shengliang Pan and Chengyang Yi. A comparison theorem with applications to sharp geo- metric inequalities for submanifolds.arXiv:2605.06074v2, 2026

work page Pith review arXiv 2026

[2] [1]

Sharp geometric inequali- ties for closed hypersurfaces in manifolds with nonnegative Ricci curvature.Invent

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

work page 2020

[3] [2]

Comparison geometry for substatic manifolds and a weighted isoperimetric inequality.arXiv:2307.14618, 2023

Stefano Borghini and Mattia Fogagnolo. Comparison geometry for substatic manifolds and a weighted isoperimetric inequality.arXiv:2307.14618, 2023

arXiv 2023

[4] [3]

S. Brendle. Geometric inequalities and the Alexandrov-Bakelman-Pucci technique.Science China Mathematics, https://doi.org/10.1007/s11425-026-2652-7, 2026

work page doi:10.1007/s11425-026-2652-7 2026

[5] [4]

Sobolev inequalities in manifolds with nonnegative curvature.Comm

Simon Brendle. Sobolev inequalities in manifolds with nonnegative curvature.Comm. Pure Appl. Math., 76(9):2192–2218, 2023

work page 2023

[6] [5]

Yousef K. Chahine. Volume estimates for tubes around submanifolds using integral curvature bounds.J. Geom. Anal., 30(4):4071–4091, 2020

work page 2020

[7] [6]

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

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

work page 1971

[8] [7]

Sobolev Inequality In Manifolds With Lower Quadratic Curvature Decay

Tian Chong, Han Luo, and Lingen Lu. Sobolev inequality in manifolds with lower quadratic curvature decay.arXiv preprint arXiv:2503.19254, 2025

work page Pith review arXiv 2025