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A scalar-mean curvature comparison theorem for manifolds with iterated conical singularities

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Pith's one-line read Scalar-mean rigidity extends to spin manifolds with iterated conical singularities.

desk verdict A serious and important extension of scalar-mean comparison to iterated conical singularities, but the spectral-gap lemma that carries the analytic core has a proof gap that needs to be fixed. read the letter →

arxiv 2506.24059 v1 pith:OKJYT5VG submitted 2025-06-30 math.DG

Milan Jovanovic , Jinmin Wang This is my paper · ORCID
classification math.DG MSC 53C2153C2758J2053C24
keywords scalar-meancurvaturecomparisoniteratedconicalsingularitiestwistedDiracoperatorFredholmnessrigiditypositivemasstheoremspinmanifolds
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 scalar-mean curvature comparison theorem for compact spin manifolds with iterated asymptotically conical singularities (IACS), spaces whose singular strata are recursively built from cones over lower-dimensional singular links. The claim is that if an asymptotically conical map between two such manifolds satisfies the comparison inequalities \(\mathrm{Sc}_{\hat g}\ge \|dF\|^2F^*\mathrm{Sc}_g\) and \(H_{\hat g}\ge \|d(F_\partial)\|\)\((F_\partial)^*H_g\), with the target having nonnegative curvature operator and nonnegative boundary second fundamental form, then both inequalities are equalities. Equality then drives rigidity: a target with positive Ricci curvature makes the map a Riemannian covering up to homothety, and a flat target forces a Ricci-flat domain. The proof runs a twisted Dirac operator with a spectral boundary condition and uses a dichotomy argument so the Fredholm index never has to be computed explicitly. The same machinery yields a rigidity theorem for Euclidean domains and a spin positive mass theorem for asymptotically flat manifolds with iterated conical singularities, so scalar-mean rigidity survives non-isolated, iterated singularities.

What carries the argument

The carrying object is the twisted Dirac operator \(D\) on the spinor bundle of \(T\hat M\oplus F^*TM\), studied with the absolute boundary condition \(\sqrt{-1}\hat c(\hat\nu)c(\nu)\$\sigma$=-\$\sigma$\) on \(\partial\hat M\). The argument first proves self-adjointness and Fredholmness: on each conical neighborhood the operator reduces to a model that is self-adjoint when the link operator \(P\) has spectral gap \(|P|\ge \tfrac12\sqrt{(n-1)(n-2)}\), and this gap follows from the scalar-curvature comparison and nonnegative curvature operator. A Hardy-type inequality and a compact Sobolev embedding, both requiring links of dimension at least two, convert self-adjointness into Fredholmness. The proof is completed by a dichotomy: if the boundary-value Dirac operator has a kernel section, the kernel section directly forces the comparison equalities; if it is invertible, an index-theoretic input on the boundary produces a harmonic section, which again forces the equalities. This avoids computing the index of the singular twisted Dirac operator.

What would settle it

Take the cylinder example described in Remark 2.4: \(\hat M=[0,1]\times $S^{1}$/\sim\) with \(\hat g=$dr^{2}$+$r^{2}$g_{$S^{1}$}\), \(M=[0,a]\times $S^{1}$/\sim\) with \(g=$dr^{2}$+$a^{{-2}}$$r^{2}$g_{$S^{1}$}\), and \(F(r,x)=(ar,x)\); for \(a>1\) this satisfies the comparison hypotheses but not the equality conclusions. Producing such a one-dimensional-link pair in a compact boundary-preserving or nonzero-degree setting would directly disprove Theorem 5.6 as stated.

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Extended reading notes

Core claim

The central result, Theorem 5.6, the precise form of Theorem 1.1, asserts that for compact spin \(C^m\)-manifolds with iterated conical singularities and smooth boundaries, links of dimension at least two, nonnegative target curvature operator, and nonnegative boundary second fundamental form, the hypotheses \(\mathrm{Sc}_{\hat g}\ge \|dF\|^2F^*\mathrm{Sc}_g\) and \(H_{\hat g}\ge \|d(F_\partial)\|\)\((F_\partial)^*H_g\) force equality in both inequalities, provided either the boundaries coincide as Riemannian manifolds or the boundary has nonzero Euler characteristic and \(F\) has nonzero degree. If \(\mathrm{Ric}_g>0\), equality forces \(\|dF\|\) to be constant and \(F:(\hat M,a\hat g)\to(M,g)\) to be a Riemannian covering map; if \(M\) is flat, then \(\hat M\) is Ricci flat. The theorem is proved first for fiberwise asymptotically conical singularities and then extended to iterated conical singularities by induction on stratum depth, with Fredholmness of the twisted Dirac operator as the load-bearing analytic step.

Load-bearing premise

The argument depends on the singular strata having codimension at least three, equivalently all links having dimension at least two; if a one-dimensional link appears, the Hardy inequality, compact embedding, and spectral-gap bound fail, and the paper itself notes a codimension-two counterexample.

Editorial extensions

If this is right

  • If Theorem 5.6 is correct, no asymptotically conical map satisfying the comparison inequalities can improve scalar curvature or mean curvature strictly; the inequalities are always saturated.
  • In the positive-Ricci target case, the map is a Riemannian covering up to a constant homothety, so the domain metric is determined by the target metric up to scale.
  • A flat target forces the domain to be Ricci flat, giving a singular-space analogue of flat rigidity.
  • Theorem 6.1 asserts that a compact spin IACS manifold with nonnegative scalar curvature and a boundary map of nonzero degree to a closed convex Euclidean hypersurface is flat under the stated boundary conditions.
  • Theorem 6.4 asserts nonnegative ADM mass for asymptotically flat spin IACS manifolds with nonnegative scalar curvature and links of dimension at least two, with zero mass forcing flatness.

Reading between the lines

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

  • The dichotomy argument suggests that the Fredholm index of the singular twisted Dirac operator is not needed for rigidity, so structurally similar comparison statements could hold for other elliptic operators once Fredholmness is established.
  • The sharpness discussion indicates that codimension-two singularities genuinely break the result; one testable direction is whether a weaker Hardy-type inequality or a different boundary condition can restore a limiting version of the theorem for codimension-two strata.
  • One could try to extend the comparison statement beyond spin by adapting the paper's estimates to minimal-surface or \(\mu\)-bubble methods, which have produced smooth low-dimensional comparison theorems without spin.
  • The positive mass theorem part suggests constructing explicit asymptotically flat IACS examples with two-dimensional links to probe whether the zero-mass flatness conclusion is sharp.
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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

3 major / 5 minor

Summary. The paper proves a scalar-mean curvature comparison theorem for compact spin manifolds with iterated asymptotically conical singularities (IACS). The main result, Theorem 5.6, asserts that if the domain and target satisfy the scalar and mean curvature comparison inequalities, the target has nonnegative curvature operator and nonnegative boundary second fundamental form, and all singular links have dimension at least two, then the two inequalities are equalities; additional rigidity conclusions hold when the target has positive Ricci curvature or is flat. The proof uses a twisted Dirac operator, a Witt-type spectral-gap estimate for the link operator to obtain Fredholmness, and a dichotomy argument that avoids explicit computation of the index. Applications include a rigidity theorem for Euclidean domains and a spin positive mass theorem for asymptotically flat IACS manifolds.

Significance. If the proof is completed, this is a substantial extension of Llarull/Lott-type comparison rigidity to spaces with iterated conical singularities. The paper has real strengths: the codimension-two counterexample in Remark 2.4 correctly identifies why the condition on links is needed; the proof has no fitted parameters or post-hoc adjustments; and the dichotomy argument is an interesting way to bypass eta-invariant computations for the index. The Hardy-type estimates and Rellich compactness for IACS spaces in Section 5 are useful technical contributions. However, several load-bearing steps are only sketched, so the current version is not yet a complete proof of the advertised theorems.

major comments (3)
  1. [Section 3.2, Lemma 3.6] The inequality chain in Lemma 3.6 reads Sc_hatL - A >= max{phi'(1)^2, phi(1)^2||df||^2} (f^*Sc_L - A)/phi(1)^2 >= ||df||^2 (f^*Sc_L - A). The second inequality is not valid for arbitrary sign of f^*Sc_L - A; it requires f^*Sc_L - A to be nonnegative. This does follow from the hypothesis that the curvature operator of the target cone is nonnegative: by Proposition 2.6, R_CL|_{wedge^2 TL} = (1/r^2)(R_L - Id), so R_L >= Id and hence Sc_L >= (n-1)(n-2). The manuscript omits this sign argument, and without it the algebra is wrong because multiplying by a larger positive factor reverses an inequality when the bracket is negative. I also note that a normalization phi(1)=1 is not needed: the factor 1/phi(1)^2 is exactly the pullback scalar curvature, and the inequality max{...}^2 >= phi(1)^2||df||^2 supplies the correct constant.
  2. [Section 3.3, after Eq. (3.4)] The Kato-Rellich perturbation step that passes from the model operator D_{\tilde V^R} to the original twisted Dirac operator is only sketched. The paper asserts that the error terms have asymptotic order o(1/gamma) or c(hat b)/gamma with c(hat b) -> 0, and that 'taking R to be very big and shrinking hat U and U to be very small' gives the required relative bound with b < 1. These assertions are load-bearing for self-adjointness, Fredholmness, and therefore all subsequent conclusions. The manuscript should provide the actual estimates, the explicit dependence on R and the neighborhood sizes, and a verification that the cited results from [5] apply uniformly to the family of metrics considered; as written this is a proof sketch rather than a proof.
  3. [Section 4, Theorem 4.1, Case (2)] In Case (2) the proof asserts that the 'classical Atiyah-Singer index theorem applied to the boundary' yields a nonzero section in the kernel of D^{\partial,\partial} when chi(partial M) != 0 and F has nonzero degree. The operator D^{\partial,\partial} is a twisted Dirac operator on S(T hat M \oplus F^*TM)|_{\partial hat M}, and the dependence of its index on the boundary map F_\partial is not written down. The paper should state the relevant index formula, identify the coefficient bundle and grading, and explain how the degree of F_\partial enters; otherwise the dichotomy in Case (2) is unsupported.
minor comments (5)
  1. [Title/Abstract] The title and abstract contain typographical artifacts ('CUR V ATURE', 'cur-vature') that should be corrected.
  2. [Section 3.2, Lemma 3.6] After defining A = (n-1)(n-2), the proof should explicitly state that f^*Sc_L - A >= 0 as a consequence of nonnegative curvature operator of the target cone, so that the inequality max{...}^2 >= ||df||^2 can be multiplied by a nonnegative quantity.
  3. [Section 3.1, Proposition 3.2] In the Cauchy-Schwarz step of the Hardy-type estimate, the bound on \int |\sigma|^2 r^{l-1} dr in terms of \|1/r \sigma\|^2 uses that r is bounded away from infinity in the conical neighborhood; this localization should be stated explicitly.
  4. [Section 5.3] The sentence 'It is not hard to see that ||d f_{\hat b_0}|| is bounded above by some constant' should be justified, for example by compactness of the link and smoothness of the family f(hat b, .); this boundedness is needed for the induction.
  5. [Section 6.1, Theorem 6.1] The sentence 'By definition, the singularities of M lie entirely within the interior. Therefore, the map F_\partial extends to a smooth map F : M -> Omega' is terse; although the extension is immediate because Omega is a ball, the collar construction should be described for completeness.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the scalar-mean comparison argument derives the rigidity conclusion from the stated hypotheses via external Dirac-operator tools, and the author self-citations are for standard techniques rather than for the load-bearing claim.

full rationale

The paper's central derivation is self-contained rather than circular. Theorem 5.6 asserts rigidity under the comparison inequalities Sc_hatg >= ||dF||^2 F*Sc_g and H_hatg >= ||d(F_delta)|| (F_delta)^* H_g; the proof uses a twisted Dirac operator whose Fredholmness is established in Section 3 (and inductively in Section 5) from the scalar-curvature comparison, nonnegative curvature operator, and codimension-at-least-three link condition, not from the desired conclusion. The key spectral-gap estimate in Lemma 3.6 derives the bound |P| >= sqrt((n-1)(n-2))/2 from the comparison inequality and curvature nonnegativity; even if the algebraic normalization in that lemma were questionable, that would be a correctness defect rather than circularity. The paper explicitly avoids computing the index and uses a dichotomy argument, so the conclusion is not hard-wired into a fitted index or an imported index formula. Citations to the authors' earlier works ([27], [28], [29], [30], [31], [32]) are used for comparison techniques, rigidity arguments, and applications; the main theorem does not reduce to any of these citations, and the decisive Fredholmness and index tools are external ([3], [5], [7], [12], [19], [20], [26]). No fitted parameter is renamed as a prediction, no ansatz is smuggled in through a self-citation, and no known result is merely relabeled. Accordingly, the circularity score is low.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters are fitted to data. The central claim rests on standard analytic and index-theoretic results plus geometric hypotheses encoded in the theorem statements. The most consequential structural assumption is the codimension-at-least-three condition on all singular strata, without which the main theorem is false. No new particles, forces, dimensions, or other invented entities are introduced.

assumptions (8)
  • standard math Spin geometry: Lichnerowicz formula and twisted Lichnerowicz identity
    Used throughout Sections 2.4 and 2.5 to derive inequalities (2.14) and (2.15) relating the twisted Dirac operator to scalar and mean curvature.
  • standard math Witt-type spectral gap criterion for self-adjointness of first-order cone operators
    Invoked in Section 3.2 via [7] to prove self-adjointness of the model cone Dirac operator when the link operator has eigenvalues at least 1/2 in absolute value.
  • standard math Kato-Rellich perturbation theorem
    Used in Sections 3.3 and 5.3 to transfer self-adjointness from model cone metrics to the actual asymptotically conical metrics after controlling error terms.
  • domain assumption Rellich-Kondrachov compactness on FACS and IACS manifolds
    Proved by the authors in Propositions 3.4 and 5.5. This compactness is needed to pass from self-adjointness to Fredholmness via compact resolvents, and it requires links of dimension at least two.
  • standard math Atiyah-Singer index theorem for the boundary Dirac operator
    Used in Case (2) of Theorem 4.1 to produce a nonzero section in the kernel of the boundary operator when the Euler characteristic of the boundary is nonzero.
  • standard math Bartnik weighted Sobolev and ADM mass framework
    Used in Lemma 6.6 and the proof of Theorem 6.4 for the asymptotically flat end, weighted Sobolev spaces, and the boundary integral computation giving ADM mass.
  • domain assumption Codimension at least three for all singular strata
    Needed for the Hardy-type inequality in Proposition 3.2, the compact embedding in Proposition 3.4, and the spectral gap in Lemma 3.6. Remark 2.4 shows the theorem fails for codimension-two singularities.
  • domain assumption Nonnegative curvature operator on the target and nonnegative boundary second fundamental form
    Used in Theorem 2.15 and Proposition 2.16 to control the twisted curvature terms and boundary terms that drive the comparison inequalities.

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Pith. "Pith review of A scalar-mean curvature comparison theorem for manifolds with iterated conical singularities." pith.science (2026). https://pith.science/paper/OKJYT5VG

@misc{pith2026250624059,
  author       = {Pith},
  title        = {Pith review of: A scalar-mean curvature comparison theorem for manifolds with iterated conical singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OKJYT5VG}},
  note         = {Machine review of arXiv:2506.24059}
}
read the original abstract

We use the Dirac operator method to prove a scalar-mean curvature comparison theorem for spin manifolds which carry iterated conical singularities. Our approach is to study the index theory of a twisted Dirac operator on such singular manifolds. A dichotomy argument is used to prove the comparison theorem without knowing precisely the index of the twisted Dirac operator. This framework also enables us to prove a rigidity theorem of Euclidean domains and a spin positive mass theorem for asymptotically flat manifolds with iterated conical singularities.

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Works this paper leans on

34 extracted references · 26 canonical work pages

  1. [5]

    Spineurs, op´ erateurs de Dirac et variations de m´ etriques.Comm

    Jean-Pierre Bourguignon and Paul Gauduchon. Spineurs, op´ erateurs de Dirac et variations de m´ etriques.Comm. Math. Phys. , 144(3):581–599, 1992

    work page 1992

  2. [1]

    The index of Dirac operators on incomplete edge spaces

    Pierre Albin and Jesse Gell-Redman. The index of Dirac operators on incomplete edge spaces. SIGMA Symmetry Integrability Geom. Methods Appl. , 12:Paper No. 089, 45, 2016

    work page 2016

  3. [2]

    Scalar curvature rigid- ity of warped product metrics

    Christian B¨ ar, Simon Brendle, Bernhard Hanke, and Yipeng Wang. Scalar curvature rigid- ity of warped product metrics. SIGMA Symmetry Integrability Geom. Methods Appl. , 20:Paper No. 035, 26, 2024

    work page 2024

  4. [3]

    The mass of an asymptotically flat manifold

    Robert Bartnik. The mass of an asymptotically flat manifold. Comm. Pure Appl. Math. , 39(5):661–693, 1986

    work page 1986

  5. [4]

    Positive scalar curvature on simply connected spin pseudomanifolds

    Boris Botvinnik, Paolo Piazza, and Jonathan Rosenberg. Positive scalar curvature on simply connected spin pseudomanifolds. J. Topol. Anal., 15(2):413–443, 2023

    work page 2023

  6. [6]

    Scalar curvature rigidity of convex polytopes

    Simon Brendle. Scalar curvature rigidity of convex polytopes. Invent. Math. , 235(2):669– 708, 2024

    work page 2024

  7. [7]

    An index theorem for first order regular singular op- erators

    Jochen Br¨ uning and Robert Seeley. An index theorem for first order regular singular op- erators. Amer. J. Math. , 110(4):659–714, 1988

    work page 1988

  8. [8]

    Lipschitz rigidity for scalar cur- vature

    Simone Cecchini, Bernhard Hanke, and Thomas Schick. Lipschitz rigidity for scalar cur- vature. arXiv:2206.11796, 2022

    arXiv 2022

Show all 34 references
  1. [9]

    Scalar curvature rigidity of the four-dimensional sphere

    Simone Cecchini, Jinmin Wang, Zhizhang Xie, and Bo Zhu. Scalar curvature rigidity of the four-dimensional sphere. arXiv:2402.12633, 2024

    2024 arXiv

  2. [10]

    Scalar and mean curvature comparison via the Dirac operator

    Simone Cecchini and Rudolf Zeidler. Scalar and mean curvature comparison via the Dirac operator. Geom. Topol., 28(3):1167–1212, 2024

    2024

  3. [11]

    Spectral geometry of singular Riemannian spaces

    Jeff Cheeger. Spectral geometry of singular Riemannian spaces. J. Differential Geom. , 18(4):575–657, 1983

    1983

  4. [12]

    The Dirac operator on spaces with conical singularities and posi- tive scalar curvatures

    Arthur Weichung Chou. The Dirac operator on spaces with conical singularities and posi- tive scalar curvatures. Trans. Amer. Math. Soc. , 289(1):1–40, 1985

    1985

  5. [13]

    Llarull’s theorem on punctured sphere with L∞ metric

    Jianchun Chu, Man-Chun Lee, and Jintian ZhuJin. Llarull’s theorem on punctured sphere with L∞ metric. arXiv:2405.19724, 2024

    2024 arXiv

  6. [14]

    Positive mass theorem for asymptotically flat manifolds with isolated conical singularities

    Xianzhe Dai, Yukai Sun, and Changliang Wang. Positive mass theorem for asymptotically flat manifolds with isolated conical singularities. arXiv:2401.07186v1, 2024

    2024 arXiv

  7. [15]

    Positive mass theorem for asymptot- ically flat spin manifolds with isolated conical singularities

    Xianzhe Dai, Yukai Sun, and Changliang Wang. Positive mass theorem for asymptot- ically flat spin manifolds with isolated conical singularities. Trans. Amer. Math. Soc. , 378(4):2617–2642, 2025

    2025

  8. [16]

    Goette and U

    S. Goette and U. Semmelmann. Scalar curvature estimates for compact symmetric spaces. Differential Geom. Appl. , 16(1):65–78, 2002

    2002

  9. [17]

    Four lectures on scalar curvature

    Misha Gromov. Four lectures on scalar curvature. In Perspectives in scalar curvature. Vol. 1, pages 1–514. World Sci. Publ., Hackensack, NJ, [2023] ©2023

    2023

  10. [18]

    Positive mass theorems on singular spaces and some applications

    Shihang He, Yuguang Shi, and Haobin Yu. Positive mass theorems on singular spaces and some applications. arXiv:2502.18000, 2025. 46 MILAN JOV ANOVIC AND JINMIN W ANG

    2025 arXiv

  11. [19]

    Blaine Lawson, Jr

    H. Blaine Lawson, Jr. and Marie-Louise Michelsohn. Spin geometry, volume 38 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1989

    1989

  12. [20]

    Deficiency indices for symmetric Dirac operators on manifolds with conic singularities

    Matthias Lesch. Deficiency indices for symmetric Dirac operators on manifolds with conic singularities. Topology, 32(3):611–623, 1993

    1993

  13. [21]

    Operators of Fuchs type, conical singularities, and asymptotic methods , volume 136 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]

    Matthias Lesch. Operators of Fuchs type, conical singularities, and asymptotic methods , volume 136 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]. B. G. Teub- ner Verlagsgesellschaft mbH, Stuttgart, 1997

    1997

  14. [22]

    Positive scalar curvature with skeleton singularities

    Chao Li and Christos Mantoulidis. Positive scalar curvature with skeleton singularities. Math. Ann., 374(1-2):99–131, 2019

    2019

  15. [23]

    Sharp estimates and the Dirac operator

    Marcelo Llarull. Sharp estimates and the Dirac operator. Math. Ann., 310(1):55–71, 1998

    1998

  16. [24]

    Index theory for scalar curvature on manifolds with boundary

    John Lott. Index theory for scalar curvature on manifolds with boundary. Proc. Amer. Math. Soc., 149(10):4451–4459, 2021

    2021

  17. [25]

    On the proof of the positive mass conjecture in general relativity

    Richard Schoen and Shing Tung Yau. On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. , 65(1):45–76, 1979

    1979

  18. [26]

    Yamabe constants of collapsing Riemannian submersions

    Manuel Streil. Yamabe constants of collapsing Riemannian submersions . PhD thesis, Uni- versit¨ at Regensburg, 2016

    2016

  19. [27]

    Scalar-mean rigidity theorem for compact manifolds with boundary

    Jinmin Wang, Zhichao Wang, and Bo Zhu. Scalar-mean rigidity theorem for compact manifolds with boundary. arXiv:2409.14503, 2024

    2024 arXiv

  20. [28]

    On gromov’s flat corner domination conjecture and stoker’s conjecture

    Jinmin Wang and Zhizhang Xie. On gromov’s flat corner domination conjecture and stoker’s conjecture. arXiv:2203.09511, 2022

    2022 arXiv

  21. [29]

    Dihedral rigidity for submanifolds of warped product manifolds

    Jinmin Wang and Zhizhang Xie. Dihedral rigidity for submanifolds of warped product manifolds. arXiv:2303.13492, 2023

    2023

  22. [30]

    Scalar curvature rigidity of spheres with subsets removed and L∞ metrics

    Jinmin Wang and Zhizhang Xie. Scalar curvature rigidity of spheres with subsets removed and L∞ metrics. arXiv:2407.21312, 2024

    2024 arXiv

  23. [31]

    Scalar curvature rigidity of degenerate warped product spaces

    Jinmin Wang and Zhizhang Xie. Scalar curvature rigidity of degenerate warped product spaces. Trans. Amer. Math. Soc. Ser. B , 12:1–37, 2025

    2025

  24. [32]

    On Gromov’s dihedral extremality and rigidity conjectures

    Jinmin Wang, Zhizhang Xie, and Guoliang Yu. On Gromov’s dihedral extremality and rigidity conjectures. arXiv:2112.01510, 2021

    2021 arXiv

  25. [33]

    A new proof of the positive energy theorem

    Edward Witten. A new proof of the positive energy theorem. Comm. Math. Phys. , 80(3):381–402, 1981

    1981

  26. [34]

    A quantitative relative index theorem and Gromov’s conjectures on positive scalar curvature

    Zhizhang Xie. A quantitative relative index theorem and Gromov’s conjectures on positive scalar curvature. J. Noncommut. Geom., 17(2):609–662, 2023. With an appendix by Jinmin Wang and Xie. (Milan Jovanovic) Texas A&M University Email address : milankj@tamu.edu (Jinmin Wang) I...

    2023

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