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Estimates of $p$-capacity for manifolds with Ricci curvature bounded from below

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Pith's one-line read This paper proves sharp upper bounds for the p-capacity of a compact domain in a complete non-compact (n+1)-manifold with Ricci curvature at least −ng, expressed purely through the boundary's mean curvature, and characterizes equality by is

desk verdict Sharp p-capacity bounds check out, but the lower-bound half leans on unpublished preprints and the rigidity proofs skip a needed step. read the letter →

arxiv 2607.16025 v1 pith:OALWLY7J submitted 2026-07-17 math.DG

Xiaoshang Jin , Zhiwei Lü , Jiabin Yin This is my paper · ORCID
classification math.DG MSC 53C2131C15
keywords p-capacityp-LaplacianRiccicurvaturemeanwarpedproductrelativecapacitycondensereigenvalueestimate
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

This paper proves sharp upper bounds for the $p$-capacity of a compact domain in a complete non-compact $(n+1)$-manifold with Ricci curvature at least $-ng$. The bound is expressed solely through the mean curvature of the boundary and matches three warped-product models: an exponential end, a hyperbolic end, and a cosh-type end, depending on whether the mean curvature equals, exceeds, or falls below $n$. Equality is rigid: it forces the exterior of the domain to be isometric to the corresponding model end, with the boundary mean curvature constant. The paper also studies scale-invariant lower bounds for relative capacity of condensers and shows a uniform positive lower bound exists exactly when the normalization exponent $\mu$ satisfies $\mu \geq 1$, for both $\mathrm{Ric} \geq 0$ and $\mathrm{Ric} \geq -n\kappa$ settings (with a diameter constraint or an exponential factor in the negative-curvature case). A sharp Euclidean lower bound for the normalized capacity is derived as a corollary.

What carries the argument

The proof rests on the normal Jacobian comparison $J(x, r) \leq (\cosh r + \frac{H(x)}{n} \sinh r)^n$ along geodesics emanating from $\partial \Omega$, which follows from the Riccati equation under $\mathrm{Ric} \geq -ng$. This inequality converts the $p$-Dirichlet energy of a radial test function into an explicit integral over $\partial \Omega$, yielding the three sharp bounds. The test functions are radial $p$-harmonic functions from the model spaces: $e^{nr/(1-p)}$ for the exponential end, $v_p(r)$ for the hyperbolic model, and $w_p(r)$ for the cosh model. Equality in the estimates reduces to equality in the Jacobian comparison and constant mean curvature, and the rigidity conclusions then follow from splitting theorems. For the lower-bound half, the inequ

What would settle it

On the warped product $(R^{n+1}, dr^2 + f(r)^2 g_{S^n})$ with $f(r) = \sinh r$ for $r \leq 1/2$ and $f(r) = C e^{-r}$ for $r \geq 1$ (the example built in Section 4), compute the first Dirichlet $p$-eigenvalue $\lambda_{1,p}$ of the annulus $\{R \leq r \leq R+D\}$ for large $R$. If $\lambda_{1,p}$ decays faster than $e^{-n(R+D)}$ as $R \to \infty$, the key eigenvalue input from the companion paper fails and the uniform lower bound (1.8) collapses.

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Extended reading notes

Core claim

On the paper's own terms: for any $p > 1$ and any compact smooth domain $\Omega$ in a complete non-compact $(n+1)$-manifold with $\mathrm{Ric} \geq -ng$, the $p$-capacity satisfies $\mathrm{Cap}_p(\Omega) \leq \left(\frac{n}{p-1}\right)^{p-1} \int_{\partial \Omega} \max\{1, H/n\}^n d\sigma$, with equality exactly when $\partial \Omega$ is connected, $H \equiv n$, and $M \setminus \Omega$ is isometric to $([0,\infty) \times \partial \Omega, dr^2 + e^{2r} g_{\partial \Omega})$. Analogous sharp estimates hold when $H > n$ (with a hyperbolic warped-product model) and when $0 \leq H < n$ (with a cosh-type model), each with its own rigidity statement. For $\mathrm{Ric} \geq 0$ and $p \in (1, n+1)$, a separate sharp bound holds with an asymptotically flat model end. On the lower-bound side, the paper introduces scale-invariant normalizations $C_{p,\mu}$ and $D_{p,\mu}$ for relative capacity and

Load-bearing premise

The lower-bound half of the paper borrows its decisive eigenvalue estimates from two unpublished companion preprints by the same authors; if those estimates fail, the optimality of $\mu \geq 1$ in (1.5), (1.7), and (1.8) loses its support.

Editorial extensions

If this is right

  • If the bounds hold, the p-capacity of a domain is controlled by its boundary mean curvature alone, with constants matching the model ends—useful for nonlinear potential theory on general manifolds.
  • The rigidity statements mean that any domain attaining equality is visibly a warped-product end, providing a geometric characterization that could be used to identify asymptotic models from capacity data.
  • The threshold µ ≥ 1 is optimal for the normalized relative capacity, separating manifolds of nonnegative Ricci curvature (where volume growth is polynomial) from those with negative lower bounds (where exponential volume growth dominates).
  • In the Ric ≥ 0 case, the sharp Euclidean lower bound for C_{p,1} gives an explicit, computable constant for condensers, with equality for concentric balls at a specific radius ratio depending on p and n.

Reading between the lines

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

  • The sharp upper bounds likely extend to boundary terms involving higher-order curvature invariants, since the Jacobian comparison only needs the Riccati equation and scalar mean curvature—higher-order terms might give refined versions of the same comparison.
  • The phase transition at µ = 1 suggests a sharp functional-analytic threshold: below it, capacity can be made arbitrarily small relative to volume and diameter, which may reflect the parabolic/hyperbolic nature of the manifold class M_κ.
  • The Ric ≥ 0 rigidity gives a global model for an asymptotically flat end without assuming scalar curvature bounds, hinting at possible connections to mass-capacity inequalities on asymptotically flat manifolds.
  • A testable numerical extension: on a warped product with nonconstant boundary mean curvature, the ratio Cap_p(Ω) divided by the right-hand side of (1.1) should lie in [0,1] and approach 1 as the mean curvature approaches a constant; computing this ratio for smooth perturbations could verify the stability of the inequality.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper proves sharp upper bounds for the absolute p-capacity of compact smooth domains in complete non-compact manifolds with Ric ≥ -ng, expressed through the boundary mean curvature. The main results are three comparison estimates (Theorem 1.3), a Ric ≥ 0 variant (Theorem 1.5), and a family of scale-invariant lower bounds for relative p-capacity with supposedly optimal normalization exponents (Theorem 1.6). The upper-bound proofs use a Jacobian comparison estimate (2.1) from the authors' related work [12], construct explicit radial test functions, and characterize equality cases as warped-product models. The lower-bound part reduces capacity to the first p-Laplace Dirichlet eigenvalue and invokes eigenvalue estimates from two unpublished preprints [7], [8] by two of the authors.

Significance. If correct, the comparison inequalities are substantial: they give sharp, curvature-dependent bounds for p-capacity in terms of boundary mean curvature alone, with rigidity forcing warped-product ends, and they identify optimal ranges of normalization parameters for a scale-invariant relative capacity. The upper-bound chain, especially equations (2.2), (2.4), (2.6) and (2.8), is carefully computed and the model computations in Example 1.2 and Remark 1.4 are consistent. The paper is less self-contained in its second half: Theorem 1.6, a central advertised result, rests on eigenvalue estimates from two unpublished preprints by the same authors, and the rigidity claims for equality contain a gap in passing from scalar Jacobian equality to the full second fundamental form. These issues are fixable but must be addressed before the claims can be considered fully supported.

major comments (2)
  1. [Section 4, Theorem 1.6, eqs. (4.1)-(4.2) and (1.5)-(1.8)] The positive lower bounds in (1.5), (1.7) and (1.8) are proved by combining (4.1) with eigenvalue estimates from [7] and [8], both unpublished 2026 preprints by two of the present authors. Neither the statements (beyond a brief inline quotation) nor the proofs are reproduced. If those estimates are false or carry additional hypotheses, the 'if μ≥1' directions of the sharp thresholds lose their proof. Please include full statements and sufficient proofs of the eigenvalue bounds, or rework the argument to rely on published results.
  2. [Section 2.2, equality cases of Theorem 1.3] In each equality case, the step from the scalar Jacobian equality J(x,r)=e^{nr} (or its analogues) to 'D²r = J'(x,r)/(nJ(x,r)) g = g' is not justified. The Riccati comparison for J gives information about the mean curvature (the trace of D²r), not the full second fundamental form. Equality in the scalar comparison does not by itself imply the matrix Riccati equality. A separate matrix comparison argument is needed to conclude that the level sets have isotropic second fundamental form and hence that the metric splits as the advertised warped product. The same gap appears in the rigidity statements for (1.2), (1.3), (1.4), and Theorem 3.1.
minor comments (3)
  1. [Section 2.2, first equality case] The connectedness of ∂Ω is argued by invoking Theorem 3 of [3]. The hypotheses of that reference (dealing with boundaries in an AdS/CFT setting) should be checked or stated explicitly; otherwise the equality characterization is incomplete.
  2. [Section 3.1, proof of Theorem 3.1] The phrase 'the same analysis in Section 3' should refer to Section 2.2; the referred analysis is in the preceding section of the current numbering.
  3. [References [7], [8]] The paper cites two 2026 preprints by two of the authors for the key eigenvalue estimates. These should either be replaced by published sources, or their precise statements and necessary hypotheses should be stated in the text, as they are not yet publicly verifiable.

Circularity Check

3 steps flagged · score 4.0 of 10

Upper-bound capacity theorems are self-contained; Theorem 1.6's optimal μ≥1 thresholds rest on unpublished same-author eigenvalue estimates [7],[8], making the lower-bound half load-bearing on self-citation.

  1. self citation load bearing [Section 4, proof of Theorem 1.6(1.5), after eq. (4.1)]
    "Combining this with the estimate in [7], λ_{1,p}(O)>(p−1)(π_p/(2diam(O)))^p & π_p=2π/(p sin π/p), we obtain that C^{(M,g)}_{p,1}(K,O)=Cap_p(K,O)·diam(O)^p/|K|>(p−1)(π_p/2)^p>0"

    The positive inf in (1.5) for μ=1 is deduced from a p-eigenvalue lower bound taken from [7], an unpublished 2026 preprint by two of the present authors (Jin & Lü); neither the statement nor its proof is reproduced. The capacity lower bound is therefore exactly as strong as [7]. If that eigenvalue estimate fails or has different hypotheses, the advertised 'if and only if μ≥1' loses its proof. This is load-bearing self-citation rather than a definitional reduction.

  2. self citation load bearing [Section 4, proof of Theorem 1.6(1.7), part (3)]
    "First let μ=1. Fix D0>0. By Corollary 1.5 in [8], every domain O with D=diam(O)≤D0 satisfies λ_{1,p}(O)≥C(p,n,κ,D0)/D^p. From (4.1), we obtain C^{(M,g)}_{p,1}(K,O)=Cap_p(K,O)D^p/|K|≥λ_{1,p}(O)D^p≥C(p,n,κ,D0)>0."

    The uniform lower bound for μ=1 under diam(O)≤D0, and hence the claimed μ≥1 threshold in (1.7), is supported entirely by Corollary 1.5 of [8], another unpublished 2026 preprint by Jin & Lü. The paper reproduces no proof or statement of that eigenvalue bound. The capacity conclusion is not self-contained; its validity is only as strong as the imported self-citation.

1 more flagged steps
  1. self citation load bearing [Section 4, proof of Theorem 1.6(1.8), part (4)]
    "If μ=1, then Corollary 1.5 in [8] and (4.1) give Cap_p(K,O)/|K| ≥ λ_{1,p}(O) ≥ λ̄_{diam(O),p,n+1} ≥ C(n,p)e^{−n diam(O)}, which is D^{(M,g)}_{p,1}(K,O)=Cap_p(K,O)e^{n diam(O)}/|K| ≥ C(n,p)>0."

    The exponential lower bound needed for the D-normalized capacity is imported from Corollary 1.5 of [8], again an unpublished preprint by two of the present authors. No derivation or verification appears in the paper. Consequently the 'if and only if μ≥1' conclusion in (1.8) rests on that self-citation chain rather than on an argument internal to this paper.

full rationale

No self-definitional or fitted-input circularity is present in the sharp upper-bound results. Theorems 1.3 and 1.5 are obtained by writing explicit radial test functions and integrating against the Jacobian comparison (2.1), which is a standard Riccati-comparison inequality; the target capacity bound is not assumed as an input, and no fitted parameter is relabeled as a prediction. The equality-case step from scalar Jacobian equality to the full second fundamental form identity is a possible technical gap, but it is a correctness issue, not a circularity. The main circularity-type concern is Theorem 1.6: its positive lower bounds (1.5), (1.7), and (1.8) are proved by combining (4.1) with p-eigenvalue estimates taken directly from unpublished preprints [7] and [8], both by Jin and Lü. The paper offers no indication that these cited results are machine-checked, reproduced, or otherwise externally verified, so they cannot be treated as independent support under the stated reviewing rules. The cited eigenvalue inequalities are distinct from the capacity conclusions, so this is load-bearing self-citation rather than a reduction by construction; hence a score of 4 rather than 6+ is appropriate.

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

The central estimates are parameter-free; no constants are fitted. The paper's real ledger is external theorems: the Jacobian comparison (2.1), p-Laplacian eigenvalue bounds [7]/[8], and splitting theorems [3]/[13]. The eigenvalue bounds are the most fragile entries because [7] and [8] are the authors' own unpublished preprints and are not reproduced here; the equality-case rigidity also leans on [3] in a way that may not match the hypotheses as stated.

assumptions (5)
  • domain assumption Jacobian comparison J(x,r) ≤ (cosh r + (H/n) sinh r)^n along distance spheres under Ric ≥ -ng (eq. (2.1))
    Load-bearing geometric input for Theorems 1.3 and 1.5; cited to Lemma 2.1 of [12] (a preview by Jin–Yin). The result is classical Heintze–Karcher-type comparison, but the paper proves neither it nor its equality conditions.
  • domain assumption λ_1,p(O) lower bounds: λ_1,p > (p-1)(π_p/(2 diam))^p for Ric ≥ 0 (from [7]); λ_1,p ≥ C(p,n,κ,D_0)/D^p and λ_1,p ≥ C(n,p) e^{-n diam(O)} for Ric ≥ -nκ (Corollary 1.5 of [8])
    These enter the proof of Theorem 1.6 through (4.1). [7] and [8] are unpublished preprints by Jin & Lü (same authors); the estimates are stated but not proven in the present paper.
  • domain assumption Theorem 3 in [3] (Cai–Galloway): a Ric ≥ -ng manifold with (a family of) H ≡ n hypersurfaces at diverging distance either has one end or splits as ℝ×Σ
    Used in §2.2 to conclude ∂Ω is connected in equality cases (1) and (2). Not stated in the paper; in case (2) the level sets have H = n coth(k+θ₀) ≠ n, so the hypothesis does not appear to be met as written.
  • domain assumption Theorem B in [13] (Kasue) + Cheeger–Gromoll splitting used to conclude ∂Ω connected in the Ric ≥ 0 equality case (Theorem 1.5)
    Cited without statement.
  • domain assumption Existence/uniqueness of the p-capacitary potential requires p-hyperbolicity of the end (Theorem 4.1 of [4], Fogagnolo–Mazzieri)
    Invoked in Definition 1.1 and used implicitly when the test function is identified with the potential in the equality analysis.

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Pith. "Pith review of Estimates of $p$-capacity for manifolds with Ricci curvature bounded from below." pith.science (2026). https://pith.science/paper/OALWLY7J

@misc{pith2026260716025,
  author       = {Pith},
  title        = {Pith review of: Estimates of $p$-capacity for manifolds with Ricci curvature bounded from below},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OALWLY7J}},
  note         = {Machine review of arXiv:2607.16025}
}
abstract

We study sharp estimates for the $p$-capacity on complete non-compact Riemannian manifolds under lower Ricci curvature bounds. First, we establish sharp comparison inequalities for the $p$-capacity of bounded smooth domains in manifolds satisfying $\operatorname{Ric}\ge -ng.$ The estimates are expressed in terms of the boundary mean curvature and correspond to natural warped-product model ends. We characterize all equality cases and show that equality forces the exterior region to be isometric to the corresponding warped product. We also obtain an analogous sharp estimate under nonnegative Ricci curvature, whose equality case is described by an asymptotically flat model end. Second, we investigate normalized lower bounds for the relative $p$-capacity of condensers. We introduce scale-invariant quantities involving the volume of the inner set and the diameter of the ambient domain, establish uniform positive lower bounds, and determine the optimal ranges of the normalization parameters.

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Reference graph

Works this paper leans on

26 extracted references · 3 linked inside Pith

  1. [7]

    Estimate for the first Dirichlet eigenvalue ofp−Laplacian on non-compact manifolds, 2026

    Xiaoshang Jin and Zhiwei Lü. Estimate for the first Dirichlet eigenvalue ofp−Laplacian on non-compact manifolds, 2026

    2026

  2. [8]

    Principalp−frequency estimates on non-compact mani- folds with negative Ricci curvature, 2026

    Xiaoshang Jin and Zhiwei Lü. Principalp−frequency estimates on non-compact mani- folds with negative Ricci curvature, 2026

    2026

  3. [3]

    Boundaries of zero scalar curvature in the ads/cft

    Mingliang Cai and Gregory J Galloway. Boundaries of zero scalar curvature in the ads/cft. Adv. Theor. Math. Phys, 6:1769–1783, 2000

    2000

  4. [12]

    Willmore-type inequality for closed hypersurfaces in com- plete manifolds with ricci curvature bounded below.arXiv preprint arXiv:2402.02465, 2024

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

    arXiv 2024

  5. [1]

    Full capacity–volumetry of sharp exp-integrability law

    David R Adams and Jie Xiao. Full capacity–volumetry of sharp exp-integrability law. Journal of the London Mathematical Society, 112(2):e70255, 2025

    2025

  6. [2]

    On the capacity of surfaces in manifolds with nonnegative scalar curvature.Inventiones mathematicae, 172(3):459–475, 2008

    Hubert Bray and Pengzi Miao. On the capacity of surfaces in manifolds with nonnegative scalar curvature.Inventiones mathematicae, 172(3):459–475, 2008

    2008

  7. [4]

    Minimising hulls, p-capacity and isoperi- metric inequality on complete riemannian manifolds.Journal of Functional Analysis, 283(9):109638, 2022

    Mattia Fogagnolo and Lorenzo Mazzieri. Minimising hulls, p-capacity and isoperi- metric inequality on complete riemannian manifolds.Journal of Functional Analysis, 283(9):109638, 2022. 20

    2022

  8. [5]

    Isoperimetric inequalities and capacities on riemannian manifolds

    Alexander Grigor’yan. Isoperimetric inequalities and capacities on riemannian manifolds. InThe Maz’ya Anniversary Collection: Volume 1: On Maz’ya’s work in functional anal- ysis, partial differential equations and applications, pages 139–153. Springer, 1999

    1999

Show all 26 references
  1. [6]

    The relative volume function and the capacity of sphere on asymptotically hyperbolic manifolds.Acta Mathematica Scientia, 45(3):755–770, 2025

    Xiaoshang Jin. The relative volume function and the capacity of sphere on asymptotically hyperbolic manifolds.Acta Mathematica Scientia, 45(3):755–770, 2025

    2025

  2. [9]

    Sharp estimates for capacities-to- quermassintegrals within hyperbolic space.Submitted, 2026

    Xiaoshang Jin, Yao Wan, and Jie Xiao. Sharp estimates for capacities-to- quermassintegrals within hyperbolic space.Submitted, 2026

    2026

  3. [10]

    Essential p-capacity-volume estimates for rotationally sym- metric manifolds.Advances in Mathematics, 482:110603, 2025

    Xiaoshang Jin and Jie Xiao. Essential p-capacity-volume estimates for rotationally sym- metric manifolds.Advances in Mathematics, 482:110603, 2025

    2025

  4. [11]

    Sharply estimating hyperbolic capacities.arXiv preprint arXiv:2502.13651, 2025

    Xiaoshang Jin and Jie Xiao. Sharply estimating hyperbolic capacities.arXiv preprint arXiv:2502.13651, 2025

    2025 arXiv

  5. [13]

    Ricci curvature, geodesics and some geometric properties of riemannian manifolds with boundary.Journal of the Mathematical Society of Japan, 35(1):117–131, 1983

    Atsushi Kasue. Ricci curvature, geodesics and some geometric properties of riemannian manifolds with boundary.Journal of the Mathematical Society of Japan, 35(1):117–131, 1983

    1983

  6. [14]

    Capacity of condensers and spatial mappings quasiconformal in the mean

    VI Kruglikov. Capacity of condensers and spatial mappings quasiconformal in the mean. Mathematics of the USSR-Sbornik, 58(1):185–205, 1987

    1987

  7. [15]

    Sharp upper bounds for the capacity in the hyperbolic and euclidean spaces.Advances in Nonlinear Analysis, 14(1):20250068, 2025

    Haizhong Li, Ruixuan Li, and Changwei Xiong. Sharp upper bounds for the capacity in the hyperbolic and euclidean spaces.Advances in Nonlinear Analysis, 14(1):20250068, 2025

    2025

  8. [16]

    Monotone quantities forp-harmonic functions and the sharpp-penrose inequality.arXiv preprint arXiv:2305.19784, 2023

    Liam Mazurowski and Xuan Yao. Monotone quantities forp-harmonic functions and the sharpp-penrose inequality.arXiv preprint arXiv:2305.19784, 2023

    2023 arXiv

  9. [17]

    Springer, 2013

    Vladimir Maz’ya.Sobolev spaces. Springer, 2013

    2013

  10. [18]

    Implications of some mass-capacity inequalities.The Journal of Geometric Analysis, 34(8):241, 2024

    Pengzi Miao. Implications of some mass-capacity inequalities.The Journal of Geometric Analysis, 34(8):241, 2024

    2024

  11. [19]

    Mass, capacitary functions, and the mass-to-capacity ratio: P

    Pengzi Miao. Mass, capacitary functions, and the mass-to-capacity ratio: P. miao.Peking Mathematical Journal, 8(2):351–404, 2025. 21

    2025

  12. [20]

    Adm mass, area and capacity in asymptotically flat 3-manifolds with nonnegative scalar curvature.Communications in Contemporary Mathematics, 27(09):2550011, 2025

    Francesca Oronzio. Adm mass, area and capacity in asymptotically flat 3-manifolds with nonnegative scalar curvature.Communications in Contemporary Mathematics, 27(09):2550011, 2025

    2025

  13. [21]

    Area, volume, and capacity in non-compact 3-manifolds with non- negative scalar curvature.International Mathematics Research Notices, 2025(18):rnaf282, 2025

    Francesca Oronzio. Area, volume, and capacity in non-compact 3-manifolds with non- negative scalar curvature.International Mathematics Research Notices, 2025(18):rnaf282, 2025

    2025

  14. [22]

    Isoperimetric inequalities in mathematical physics.(am-27), 1951

    G Polya and G Szegö. Isoperimetric inequalities in mathematical physics.(am-27), 1951

    1951

  15. [23]

    New monotonicity for p-capacitary functions in 3-manifolds with nonnegative scalar curvature.Advances in Mathematics, 440:109526, 2024

    Chao Xia, Jiabin Yin, and Xingjian Zhou. New monotonicity for p-capacitary functions in 3-manifolds with nonnegative scalar curvature.Advances in Mathematics, 440:109526, 2024

    2024

  16. [24]

    The p-harmonic capacity of an asymptotically flat 3-manifold with non-negative scalar curvature

    Jie Xiao. The p-harmonic capacity of an asymptotically flat 3-manifold with non-negative scalar curvature. InAnnales Henri Poincaré, volume 17, pages 2265–2283. Springer, 2016

    2016

  17. [25]

    P-capacity vs surface-area.Advances in Mathematics, 308:1318–1336, 2017

    Jie Xiao. P-capacity vs surface-area.Advances in Mathematics, 308:1318–1336, 2017

    2017

  18. [26]

    Sharp upper bounds of p-capacity in convex cone and asymptotically flat half-space.Letters in Mathematical Physics, 116(2):34, 2026

    Jiabin Yin and Xingjian Zhou. Sharp upper bounds of p-capacity in convex cone and asymptotically flat half-space.Letters in Mathematical Physics, 116(2):34, 2026. Xiaoshang Jin School of mathematics and statistics, Huazhong University of science and technology, 430074, Wuhan, ...

    2026

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