Gap phenomenon for scalar curvature

Changliang Wang; Yukai Sun

REVIEW 3 major objections 5 minor 29 references

Gap phenomenon for scalar curvature

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves a scalar-curvature gap estimate with no sign restriction on the curvature operator, and uses it to show every metric on a closed even-dimensional manifold with nonzero Euler characteristic is ε-gap distance extremal.

desk verdict The closed-manifold scalar curvature estimate is a genuine and likely correct advance; the boundary theorem relies on an unverified borrowed boundary condition that needs real work. read the letter →

arxiv 2501.01252 v1 pith:W622NO52 submitted 2025-01-02 math.DG

Yukai Sun , Changliang Wang This is my paper · ORCID

classification math.DG MSC 53C2153C2758J20

keywords scalarcurvatureepsilon-gapextremalityDiracoperatorspinmanifoldEulercharacteristicmeanfill-inproblem

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 establishes a quantitative upper bound on how much scalar curvature can rise when a metric is enlarged on an even-dimensional closed manifold with nonzero Euler characteristic. The main estimate says that, for closed spin manifolds $(M^{2n}, g_0)$ and $(N^{2n}, g)$ linked by an area-nonincreasing map $f: N \to M$ of nonzero degree, the infimum of $\mathrm{Sc}_g - \mathrm{Sc}_{g_0}(f)$ is at most $-2n(2n-1)R_{\min} + 2n(2n-1)|R_{\min}|$, where $R_{\min}$ is the smallest eigenvalue of the curvature operator of $g_0$. Previous estimates of this type required the curvature operator to be nonnegative; the new argument removes that assumption. Taking $f$ to be the identity shows that every complete metric on such a manifold is $\epsilon$-gap distance extremal for $\epsilon = 2n(2n-1)(|R_{\min}|-R_{\min})$. A boundary version compares scalar curvature and mean curvature, and yields an obstruction to extending boundary metrics with prescribed lower bounds into Euclidean domains.

What carries the argument

The twisted Dirac operator $D_E$ on the bundle $S(N) \otimes f^*S(M)$, together with the Lichnerowicz-Weitzenböck-Bochner formula, is the load-bearing object. The new step is to write the curvature operator $R$ of the target as $(R - R_{\min}) + R_{\min}$, with $R_{\min}$ the minimum eigenvalue, so that $R - R_{\min}$ is nonnegative and admits a square root $L$; Clifford algebra estimates on $L$ then yield a lower bound on the curvature term without any sign assumption on $R$. In the boundary theorem the same split is applied to the second fundamental form $A^M$, with its minimum eigenvalue $A^M_{\min}$, and a boundary condition borrowed from the cited literature makes the boundary spinor term nonnegative.

What would settle it

A counterexample would be a pair of closed spin manifolds $(M^{2n},g_0)$ and $(N^{2n},g)$ with $\chi(M)\neq 0$ and an area-nonincreasing map $f:N\to M$ of nonzero degree for which $\inf_x [\mathrm{Sc}_g(x)-\mathrm{Sc}_{g_0}(f(x))]$ is strictly greater than $-2n(2n-1)R_{\min}+2n(2n-1)|R_{\min}|$; such a pair would refute Theorem 3. The product $S^2\times \Sigma$ with varying hyperbolic surface factors is a concrete family in which the bound can be checked numerically, and the paper's Example 5 shows equality can be approached by taking the surface curvature to zero.

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Extended reading notes

Core claim

The central claim is Theorem 3: under the hypotheses above, $\inf_x [\mathrm{Sc}_g(x) - \mathrm{Sc}_{g_0}(f(x))] \leq -2n(2n-1)R_{\min} + 2n(2n-1)|R_{\min}|$, and equality forces rigidity, namely $R_{\min} \geq 0$ and $\mathrm{Sc}_g(x) = \mathrm{Sc}_{g_0}(f(x))$. The proof runs through a twisted Dirac operator on $N$ with coefficients in the pullback spinor bundle, and the new ingredient is an estimate of the curvature term in the Lichnerowicz-Weitzenböck-Bochner formula that subtracts the minimum eigenvalue $R_{\min}$ and controls the remainder with Clifford algebra inequalities. Because the index of the twisted Dirac operator is nonzero, being equal to $\deg(f)\chi(M)$, a harmonic spinor must exist, forcing the scalar-curvature gap. The identity-map case gives Theorem 2, the $\epsilon$-gap distance extremality of every metric; the boundary case, Theorem 6, gives a similar alternative between a scalar-curvature gap in the interior and a mean-curvature gap on the boundary, with an application to Euclidean domains in Corollary 7.

Load-bearing premise

The boundary theorem depends on the existence of a boundary condition, taken from the cited literature, that makes a boundary spinor integral nonnegative and the twisted Dirac operator have nonzero index, and the paper does not prove that this condition remains available after dropping the nonnegativity assumptions on the curvature operator and the second fundamental form.

Editorial extensions

If this is right

On any closed even-dimensional manifold with nonzero Euler characteristic, every complete metric is $\epsilon$-gap distance extremal for $\epsilon = 2n(2n-1)(|R_{\min}|-R_{\min})$, so no larger metric can raise the scalar curvature by more than $\epsilon$.
If the scalar-curvature gap bound is attained with equality, the metric comparison is rigid: the map is an isometry, $R_{\min} \geq 0$, and the scalar curvatures agree, so the base metric is distance rigid in that case.
The estimate extends sphere-target and nonnegative-curvature-operator results to arbitrary curvature operators, interpolating between the nonnegative and nonpositive cases listed in the paper's table.
For a Euclidean domain with nonzero Euler characteristic, the boundary version rules out any metric with nonnegative scalar curvature that agrees with the Euclidean metric on the boundary while making the boundary mean curvature exceed a computable threshold.
The same theorem covers maps between different manifolds, so it constrains scalar-curvature gaps not only for identity maps but for all area-nonincreasing maps of nonzero degree onto such targets.

Reading between the lines

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

A natural next step is to run the same minimum-eigenvalue subtraction through other Dirac-type operators; the signature operator, for instance, could yield gap bounds for metric variations of the curvature tensor, which the paper does not discuss.
Because the gap $\epsilon$ grows with $|R_{\min}|$, the extremality statement is strongest when the base metric has nonnegative curvature operator; this suggests that the new theorem is best viewed as the correct replacement for the old zero-gap result in the presence of negative curvature, not as a sharp rigidity statement.
The boundary theorem could be tested by checking whether the cited boundary condition can be constructed for arbitrary second fundamental forms; if it cannot, Corollary 7 would require an extra hypothesis on the boundary and the non-existence statement would be narrower than stated.

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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

Dear [Colleague],

The thing to know about this paper: the closed-manifold core is a genuine and likely correct advance, while the boundary theorem is asserted on top of an unverified borrowed boundary condition. If you read only one thing, read Section 2.

What's new: Sun and Wang remove the nonnegative curvature operator assumption that was essential in Goette-Semmelmann and Lott. The trick is to split the target curvature operator as R = (R - Rmin) + Rmin and estimate both parts separately, using area nonincreasingness to control the Rmin term. The resulting inequality (Theorem 3) gives an explicit constant depending on Rmin. This is not a repackaging; I don't see it in earlier literature. The application to epsilon-gap extremality for every even closed manifold with nonzero Euler characteristic is a natural and attractive consequence. The index identity Ind(D_E) = deg(f)chi(M) is standard.

I checked the algebra in (8) and (9) and it seems right. The proofs follow Llarull and Goette-Semmelmann closely; the square trick for the nonnegative operator R - Rmin is standard and correctly applied. The count of no free parameters is accurate.

Soft spots, in order of importance:

1. Theorem 6 (boundary case). After equation (11), the proof says 'we impose the boundary condition as in section 2.2 in [19] or section 3.1 in [26] such that int >= 0 and Ind(D_E) != 0.' That is doing a lot of work. The paper has just discarded Lott's nonnegativity assumptions, but gives no argument that the boundary condition still delivers those two properties. This is not a minor gap: Theorem 6, Corollary 7, and the remarks on Gromov's Question 8 all rest on it. It may be fixable by importing Lott's actual boundary analysis, but as written it's an assertion, not a proof.

2. The equality-case rigidity in Theorem 3 is sketched. In dimension 2, lambda_i lambda_j = 1 for i != j only gives lambda_1 lambda_2 = 1, not that both equal 1. So the claim that f is an isometry in the equality case may need a separate argument for surfaces. This doesn't affect the main estimate, but it should be fixed.

3. Example 5 claims the estimate is optimal, but the calculation only shows both sides tend to 0 as c -> 0. That does not establish optimality of the coefficient. The example is harmless, but the claim is stronger than the evidence.

Bottom line: the closed-manifold gap theorem is a solid, useful result and deserves a serious referee. The boundary part needs real work before it can be taken as stated. I'd send it to review with a request that the referee focus on Theorem 6's boundary condition and the dimension-2 rigidity detail.

Best, [You]

Referee Report

3 major / 5 minor

Summary. The paper derives scalar curvature comparison estimates for spin manifolds without imposing nonnegativity of the curvature operator. For closed manifolds, Theorem 3 gives an upper bound for inf_x(Sc_g(x)-Sc_{g0}(f(x))) in terms of the minimum eigenvalue Rmin of the curvature operator of the target metric, assuming an area-nonincreasing map of nonzero degree to a manifold with nonzero Euler characteristic. Theorem 2 converts this into an epsilon-gap distance extremality statement for any metric on a closed even-dimensional manifold with nonzero Euler characteristic. For compact manifolds with boundary, Theorem 6 states a similar interior-or-boundary alternative involving scalar curvature and mean curvature, and Corollary 7 applies this to Euclidean domains to address a question of Gromov. The proof of the closed-manifold theorem follows the Goette-Semmelmann Lichnerowicz-Weitzenbock-Bochner argument with a splitting of the curvature operator into Rmin plus a nonnegative part. The boundary theorem follows Lott and Wang-Xie-Yu but relies on a boundary condition whose availability is not proved in the present generality.

Significance. If the closed-manifold theorem is correct, it is a useful extension of Llarull and Goette-Semmelmann results: it removes the nonnegativity assumption on the target curvature operator and produces an explicit, parameter-free constant in the scalar curvature estimate. The application to epsilon-gap distance extremality in Theorem 2 is a clean and interesting consequence. The proof of Theorem 3 is essentially a direct Bochner-type calculation, and the index-theoretic input is standard. The boundary results, if made rigorous, would address a concrete aspect of Gromov's extension question. However, the boundary part currently contains unproved assumptions, and the Corollary 7 application does not verify the hypotheses of Theorem 6 as stated.

major comments (3)

[Section 3, paragraph after Eq. (11)] The proof of Theorem 6 invokes, without proof, a boundary condition from section 2.2 of [19] or section 3.1 of [26] that is required to satisfy both ∫∂N⟨φ,D∂Nφ⟩≥0 and Ind(DE)≠0. The present setting has removed the nonnegativity assumptions on the curvature operator and the second fundamental form, while the cited constructions may depend on such assumptions to control the boundary Dirac term and to identify the index. Since the Atiyah-Patodi-Singer formula contains boundary eta contributions, Ind(DE) is not automatically equal to deg(f)χ(M) once the boundary condition is changed. This is a load-bearing point: the disjunction (4) or (5), Corollary 7, and the discussion of Question 8 all depend on it. The authors should state the boundary condition as a lemma and prove that it exists in the present generality, or cite precisely the theorem that supplies it under the stated hypotheses.
[Corollary 7] Corollary 7 is stated as an application of Theorem 6 with f the identity map on a Euclidean domain, but the hypotheses of Theorem 6 are not verified. For f:(Ω,g)→(Ω,g_E), the area-nonincreasing condition requires |v∧w|_{g_E}≤|v∧w|_g for all tangent bivectors; this does not follow from Sc_g≥0 and is not implied by the definition of ĉ, which only controls tangent vectors on the boundary. The appearance of ĉ in the boundary inequality suggests that a variant of Theorem 6 with a conformally scaled target metric or a rescaled map is being used, but such an argument is not supplied. As written, Corollary 7 does not follow from Theorem 6 and needs either additional hypotheses or a separate proof.
[Theorem 6, equality case] In the rigidity part of Theorem 6, the conclusion '∂f:∂N→∂M is local isometric if AM_min≠0' is asserted after saying all inequalities become equalities. The tracking of equalities is only sketched, and it is not shown that the equalities force the pointwise eigenvalue condition that makes ∂f a local isometry. Since this rigidity statement is not used in the application to Corollary 7, the authors could either supply the missing details or clearly mark this part as a secondary claim.

minor comments (5)

[Remark 4] There is a typo: 'Stierel-Whitney c1ass' should be 'Stiefel-Whitney class'.
[Theorem 6 statement] For manifolds with boundary, the degree of f is not explicitly defined; the authors should specify that f is orientation-preserving with respect to the boundary orientations and that deg(f) is the relative degree in H_{2n}(N,∂N).
[Equation (14) and surrounding text] The notation AM_min(g0)αi is used without defining the subscript notation; it would be clearer to write the components of the operator AM - AM_min g0 explicitly.
[Section 3, proof of inequality (13)] The symbol f^*HM appears where (∂f)^*HM would be more precise, since HM is a function on ∂M and the pullback is along ∂f.
[Theorem 2] The statement says 'a complete Riemannian metric g0 on M' although M is closed; completeness is automatic, so the wording is redundant but not harmful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main estimates are derived from the Dirac operator, Lichnerowicz-Weitzenböck-Bochner identity, and Atiyah–Singer index theory with explicit geometric constants, not from fitted inputs or self-referential uniqueness claims.

full rationale

The paper's central derivation chain is self-contained in the relevant sense. Theorem 3 is proved by applying the Lichnerowicz-Weitzenböck-Bochner formula to a twisted Dirac operator, splitting the target curvature operator as R = (R - R_min) + R_min, and using the assumption that f is area nonincreasing to control the resulting terms. The constants R_min and A_min are defined directly from the given metrics g0 and g, and they enter the conclusion as explicit quantities rather than as parameters fitted to the target inequality. The index identity Ind(D_E) = deg(f) chi(M) is cited to Goette-Semmelmann's proof of the standard Atiyah-Singer computation for twisted Dirac operators, which is external mathematical content, not a claim established by the present paper. Theorem 2 is a direct corollary obtained by taking N = M and f = identity, so it inherits the same derivation. For the boundary theorem, Theorem 6, the proof imports a boundary condition from Lott [19] and Wang-Xie-Yu [26] to guarantee nonnegativity of the boundary Dirac term and nonvanishing index. These cited works are not by the present authors, so this is not a self-citation chain. The skeptic's concern that the cited boundary condition may not remain available after removing nonnegativity assumptions is a potential correctness gap or missing justification in the boundary case, but it is not circularity: the paper does not define the boundary condition in terms of, or fit it to, the conclusion it is used to prove. No fitted quantity is renamed as a prediction, no known result is merely relabeled, and no uniqueness theorem from the authors' own prior work is invoked to force the choice. The derivation is a genuine application of standard index-theoretic machinery with explicit estimates. Therefore the correct circularity finding is no significant circularity, score 0.

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

The paper introduces no free parameters or invented entities. The only inputs are the geometric invariants Rmin and A_min, defined from the given metrics, and standard index-theoretic facts. The axioms list the standard tools plus two domain assumptions: area nonincreasing map and the borrowed boundary condition.

assumptions (4)

standard math Atiyah-Singer index theorem gives Ind(D_E) = deg(f) chi(M) for the twisted Dirac operator on closed manifolds.
Used in the proof of Theorem 3 to guarantee a nonzero harmonic spinor when deg(f)chi(M) != 0; cited with reference [9].
standard math Lichnerowicz-Weitzenbock-Bochner formula for twisted Dirac operator (equation (6)).
Fundamental identity connecting D^2, connection Laplacian, scalar curvature, and curvature term.
domain assumption The map f is area nonincreasing, giving lambda_i lambda_j <= 1 for the singular values of f_*.
Used in Section 2 to bound the (R-Rmin) curvature term; without it the estimate (9) fails.
domain assumption Existence of a boundary condition (from [19], [26]) giving Index != 0 and nonnegative boundary Dirac term in Theorem 6.
The paper cites this rather than proving it; it is load-bearing for the boundary estimate.

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Cite this review

Pith. "Pith review of Gap phenomenon for scalar curvature." pith.science (2026). https://pith.science/paper/W622NO52

@misc{pith2026250101252,
  author       = {Pith},
  title        = {Pith review of: Gap phenomenon for scalar curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W622NO52}},
  note         = {Machine review of arXiv:2501.01252}
}

abstract

Inspired by Goette-Semmelmann \cite{GSSU2002}, we derive an estimate for the scalar curvature without a nonnegativity assumption on curvature operator. As an application, we show that, on an even dimensional closed manifold with nonzero Euler characteristic, any Riemannian metric $g$ is $\epsilon$-gap distance extremal for some $\epsilon \geq 0$. For manifolds with boundary, inspired by Lott \cite{JL2021}, we obtained a similar estimate for scalar curvature and mean curvature. We apply the estimate on certain Euclidean domains to study a Gromov's question in \cite{GM20233} concerning the extension problem of metric on the boundary to the interior.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

29 extracted references · 26 canonical work pages

[19]

Index theory for scalar curvature on manifol ds with boundary

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

work page 2021
[26]

On gromov’s dihedral extremality and rigidity conjectures

Jinmin W ang, Zhizhang Xie, and Guoliang Yu. On gromov’s dihedral extremality and rigidity conjectures. arXiv: 2112.01510 , 2023

arXiv 2023
[1]

Scalar curvature rigidity of warped product metrics

Christian B¨ ar, Simon Brendle, Bernhard Hanke, and Yipe ng W ang. Scalar curvature rigidity of warped product metrics. SIGMA Symmetry Integrability Geom. Methods Appl. , 20:Paper No. 035, 26, 2024

work page 2024
[2]

Bettiol and McFeely Jackson Goodman

Renato G. Bettiol and McFeely Jackson Goodman. Extremal ity and rigidity for scalar cur- vature in dimension four. Selecta Math. (N.S.) , 30(1):Paper No. 7, 29, 2024

work page 2024
[3]

Scalar curvature rigidity of convex poly topes

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

work page 2024
[4]

Rigid ity of spin ﬁll-ins with non-negative scalar curvature

Simone Cecchini, Sven Hirsch, and Rudolf Zeidler. Rigid ity of spin ﬁll-ins with non-negative scalar curvature. arXiv: 2404.17533 , 2024

work page arXiv 2024
[5]

Enlargeable metrics on nonspin manifolds

Simone Cecchini and Thomas Schick. Enlargeable metrics on nonspin manifolds. Proc. Amer. Math. Soc. , 149(5):2199–2211, 2021

work page 2021
[6]

S calar curvature rigidity of the four-dimensional sphere

Simone Cecchini, Jinmin W ang, Zhizhang Xie, and Bo Zhu. S calar curvature rigidity of the four-dimensional sphere. arXiv: 2402.12633 , 2024

arXiv 2024

Show all 29 references

[7]

Scalar curvature rigidi ty of domains in a warped product

Xiaoxiang Chai and Xueyuan W an. Scalar curvature rigidi ty of domains in a warped product. arXiv: 2407.10212 , 2024

2024 arXiv
[8]

Exten sion of a theorem of Shi and Tam

Michael Eichmair, Pengzi Miao, and Xiaodong W ang. Exten sion of a theorem of Shi and Tam. Calc. Var. Partial Diﬀerential Equations , 43(1-2):45–56, 2012

2012
[9]

Goette and U

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

2002
[10]

M. Gromov. Large Riemannian manifolds. In Curvature and topology of Riemannian mani- folds (Katata, 1985) , volume 1201 of Lecture Notes in Math. , pages 108–121. Springer, Berlin, 1986

1985
[11]

Blaine Lawson, Jr

Mikhael Gromov and H. Blaine Lawson, Jr. Positive scala r curvature and the Dirac operator on complete Riemannian manifolds. Inst. Hautes ´Etudes Sci. Publ. Math. , (58):83–196, 1983

1983
[12]

A dozen problems, questions and conjectu res about positive scalar curva- ture

Misha Gromov. A dozen problems, questions and conjectu res about positive scalar curva- ture. In Foundations of mathematics and physics one century after Hi lbert, pages 135–158. Springer, Cham, 2018

2018
[13]

Four lectures on scalar curvature

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

2023
[14]

Llarull type the orems on complete manifolds with positive scalar curvature

Tianze Hao, Yuguang Shi, and Yukai Sun. Llarull type the orems on complete manifolds with positive scalar curvature. Trans. Amer. Math. Soc. , Jul. 17 2024

2024
[15]

Rigidity of 3D spher ical caps via µ-bubbles

Yuhao Hu, Peng Liu, and Yuguang Shi. Rigidity of 3D spher ical caps via µ-bubbles. Paciﬁc J. Math. , 323(1):89–114, 2023

2023
[16]

W. Kramer. The scalar curvature on totally geodesic ﬁbe rings. Ann. Global Anal. Geom. , 18(6):589–600, 2000. GAP PHENOMENON FOR SCALAR CUR V ATURE 13

2000
[17]

The dihedral rigidity conjecture for n-prisms

Chao Li. The dihedral rigidity conjecture for n-prisms. J. Diﬀerential Geom. , 126(1):329–361, 2024

2024
[18]

Sharp estimates and the Dirac operato r

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

1998
[20]

Scalar and mean curvature comparison via µ-bubbles

Daniel R¨ ade. Scalar and mean curvature comparison via µ-bubbles. Calc. Var. Partial Dif- ferential Equations , 62(7):Paper No. 187, 39, 2023

2023
[21]

Positive mass theorem and the boundary behaviors of com- pact manifolds with nonnegative scalar curvature

Yuguang Shi and Luen-Fai Tam. Positive mass theorem and the boundary behaviors of com- pact manifolds with nonnegative scalar curvature. J. Diﬀerential Geom. , 62(1):79–125, 2002

2002
[22]

Total mean c urvature of the boundary and nonnegative scalar curvature ﬁll-ins

Yuguang Shi, W enlong W ang, and Guodong W ei. Total mean c urvature of the boundary and nonnegative scalar curvature ﬁll-ins. J. Reine Angew. Math. , 784:215–250, 2022

2022
[23]

On the ﬁll-in of nonnegative scalar curvature metrics

Yuguang Shi, W enlong W ang, Guodong W ei, and Jintian Zhu . On the ﬁll-in of nonnegative scalar curvature metrics. Math. Ann. , 379(1-2):235–270, 2021

2021
[24]

Scalar curvature rigidity of paraboli cally convex domains in hyperbolic spaces

Chengzhang Sun. Scalar curvature rigidity of paraboli cally convex domains in hyperbolic spaces. arXiv: 2411.09290 , 2024

2024 arXiv
[25]

Scalar curvature rigidit y of degenerate warped product spaces

Jinmin W ang and Zhizhang Xie. Scalar curvature rigidit y of degenerate warped product spaces. arXiv: 2306.05413 , 2023

2023 arXiv
[27]

A proof of gr omov’s cube inequality on scalar curvature

Jinmin W ang, Zhizhang Xie, and Guoliang Yu. A proof of gr omov’s cube inequality on scalar curvature. arXiv:2105.12054, 2023

2023 arXiv
[28]

Nonnegative scalar curvature and area d ecreasing maps

W eiping Zhang. Nonnegative scalar curvature and area d ecreasing maps. SIGMA Symmetry Integrability Geom. Methods Appl. , 16:Paper No. 033, 7, 2020

2020
[29]

Rigidity of area-minimizing 2-spheres in n-manifolds with positive scalar curva- ture

Jintian Zhu. Rigidity of area-minimizing 2-spheres in n-manifolds with positive scalar curva- ture. Proc. Amer. Math. Soc. , 148(8):3479–3489, 2020. Key Laboratory of Pure and Applied Mathematics, School of Mat hematical Sci- ences, Peking University, Beijing, 100871, P. R. Ch...

2020

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [19]

Index theory for scalar curvature on manifol ds with boundary

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

work page 2021

[2] [26]

On gromov’s dihedral extremality and rigidity conjectures

Jinmin W ang, Zhizhang Xie, and Guoliang Yu. On gromov’s dihedral extremality and rigidity conjectures. arXiv: 2112.01510 , 2023

arXiv 2023

[3] [1]

Scalar curvature rigidity of warped product metrics

Christian B¨ ar, Simon Brendle, Bernhard Hanke, and Yipe ng W ang. Scalar curvature rigidity of warped product metrics. SIGMA Symmetry Integrability Geom. Methods Appl. , 20:Paper No. 035, 26, 2024

work page 2024

[4] [2]

Bettiol and McFeely Jackson Goodman

Renato G. Bettiol and McFeely Jackson Goodman. Extremal ity and rigidity for scalar cur- vature in dimension four. Selecta Math. (N.S.) , 30(1):Paper No. 7, 29, 2024

work page 2024

[5] [3]

Scalar curvature rigidity of convex poly topes

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

work page 2024

[6] [4]

Rigid ity of spin ﬁll-ins with non-negative scalar curvature

Simone Cecchini, Sven Hirsch, and Rudolf Zeidler. Rigid ity of spin ﬁll-ins with non-negative scalar curvature. arXiv: 2404.17533 , 2024

work page arXiv 2024

[7] [5]

Enlargeable metrics on nonspin manifolds

Simone Cecchini and Thomas Schick. Enlargeable metrics on nonspin manifolds. Proc. Amer. Math. Soc. , 149(5):2199–2211, 2021

work page 2021

[8] [6]

S calar curvature rigidity of the four-dimensional sphere

Simone Cecchini, Jinmin W ang, Zhizhang Xie, and Bo Zhu. S calar curvature rigidity of the four-dimensional sphere. arXiv: 2402.12633 , 2024

arXiv 2024