arxiv: 2604.14087 · v1 · submitted 2026-04-15 · 🧮 math.DG · math.AP

Recognition: unknown

Quantification of C⁰ Convergence in Dimension Three

Liam Mazurowski , Xuan Yao

Authors on Pith no claims yet

Pith reviewed 2026-05-10 11:45 UTC · model grok-4.3

classification 🧮 math.DG math.AP

keywords scalar curvatureC0 convergencequantitative estimatesharmonic functionsGromov conjectureelliptic estimatesthree-dimensional geometry

0 comments

The pith

In three dimensions, the infimum of scalar curvature for a metric g is at most the value of R at the origin for g0 plus C times the square root of their C0 distance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper quantifies the stability of scalar curvature lower bounds when Riemannian metrics on the unit ball in R^3 converge in the C0 topology. It shows that if g is sufficiently close to a fixed smooth metric g0 in C0 norm, then the lowest scalar curvature value attained by g cannot drop below R_g0 at the origin by more than a multiple of the square root of that C0 distance. The argument proceeds by introducing a local functional built from harmonic functions that registers the scalar curvature, then applying standard elliptic estimates to prove this functional changes continuously under C0-small metric changes. Examples demonstrate that the square-root exponent cannot be replaced by any better power in general.

Core claim

For smooth metrics g and g0 on the unit ball B in R^3, constants C and epsilon0 exist depending only on g0 such that whenever the C0 norm of g minus g0 is at most epsilon0, the inequality inf over B of R_g is less than or equal to R_g0 at the origin plus C times the square root of that C0 norm. The same statement holds with a weaker rate when g0 is only C2, while rotational symmetry of g0 improves the rate to linear. Counterexamples show the exponent 1/2 is optimal.

What carries the argument

A local quantity built from harmonic functions on the metric that detects scalar curvature, whose variation is controlled by classical elliptic PDE estimates under C0 perturbations.

If this is right

Scalar curvature lower bounds are preserved under C0 limits of metrics in dimension three.
The quantitative rate is Hölder with exponent exactly 1/2 in the general case.
For rotationally symmetric g0 the bound improves to a linear rate in the C0 distance.
The same harmonic-function method yields a partial positive answer to the question of preservation of scalar curvature bounds under convergence in measure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The harmonic-function construction may adapt to give quantitative control on other curvature functionals in low dimensions.
The result indicates that scalar curvature behaves like a 1/2-Hölder continuous quantity on the space of C0 metrics.
Numerical checks on explicit perturbations of the flat or round metric could estimate the size of the constant C in concrete cases.

Load-bearing premise

The metrics are smooth enough that harmonic functions exist and classical elliptic estimates apply to the curvature-detecting quantity.

What would settle it

A sequence of smooth metrics g_n on the ball with C0 distance to g0 equal to 1/n yet with inf R_g_n less than R_g0(0) minus n would disprove the claimed rate.

read the original abstract

We address Gromov's Quantification of $C^0$ Convergence Conjecture in dimension three. Let $B$ be the unit ball in $\mathbb R^3$. Let $g$ and $g_0$ be smooth metrics on $B$. We prove there are constants $C$ and $\epsilon_0$ depending only on $g_0$ so that \[ \inf_{x\in B} R_g(x) \leq R_{g_0}(0) + C \|g-g_0\|_{C^0}^{1/2} \] provided $\|g-g_0\|_{C^0}\leq \epsilon_0$. We also construct examples to show that the exponent $1/2$ is sharp. This explicitly quantifies the fact that scalar curvature lower bounds are preserved under $C^0$ convergence of metrics. When $g_0$ is merely $C^2$ we prove a related estimate with a slightly weaker rate, and when $g_0$ has rotational symmetry we prove a related estimate with a stronger linear rate. To prove these results, we use harmonic functions to define a local quantity that detects the scalar curvature. Then we use classical elliptic PDE estimates to show that this quantity is stable under $C^0$ perturbations of the metric. As a further application of this method, we give a partial answer to a question of Gromov on the preservation of scalar curvature lower bounds for metrics that are converging in measure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Gives the first explicit 1/2-rate bound on scalar curvature under C0 perturbations in 3D with sharpness examples, but the claimed uniformity of elliptic constants on g0 alone is not obviously justified by C0 closeness.

read the letter

The paper supplies an explicit quantitative version of the fact that lower bounds on scalar curvature are preserved under C0 convergence of metrics in dimension three. The main result is the inequality inf R_g ≤ R_g0(0) + C ||g - g0||_C0^{1/2} for small C0 distance, with C and epsilon_0 depending only on g0, plus examples showing the exponent is sharp. They also treat the C2 case with a weaker rate and the rotationally symmetric case with a linear rate, and give a partial result on measure convergence. The approach defines a local quantity via harmonic functions on each metric that detects the scalar curvature, then applies elliptic estimates to control its variation. This is new relative to the Gromov conjecture mentioned, and the sharpness constructions are concrete and useful. The method itself looks like a reasonable way to turn curvature information into a stable quantity. The soft spot is the uniformity of the constants. Standard comparison or Schauder estimates for the Laplace-Beltrami operators on g and g0 produce factors that depend on C2 or C1,1 norms of the coefficients. Mere C0 closeness does not bound those norms; one can add high-frequency oscillations of small amplitude that make derivatives arbitrarily large. The abstract claims the constants depend only on g0, so the paper must either derive a different estimate that avoids this dependence or use some approximation argument to restore uniformity. Without seeing the details it is hard to tell whether that step holds. This is aimed at people working on weak convergence of Riemannian metrics and curvature bounds in low dimensions. A reader already thinking about Gromov-type questions or stability under weak limits would find the local quantity and the examples worth looking at. It deserves a serious referee because the conjecture is standard, the quantitative claim is new, and the examples are checkable, even if the elliptic step needs verification on the constants.

Referee Report

1 major / 1 minor

Summary. The paper addresses Gromov's quantification of C^0 convergence conjecture in dimension 3. For smooth metrics g and g0 on the unit ball B in R^3, it proves the existence of C and ε0 depending only on g0 such that inf_{x in B} R_g(x) ≤ R_{g0}(0) + C ||g - g0||_{C^0}^{1/2} whenever ||g - g0||_{C^0} ≤ ε0. Sharpness of the 1/2 exponent is shown by explicit examples. Variants are proved when g0 is only C^2 (weaker rate) and when g0 is rotationally symmetric (linear rate). The proof defines a local quantity Q_g via solutions to the Laplace-Beltrami equation Δ_g u = 0 with g-independent boundary data, chosen so that Q_g detects R_g at interior points, then applies classical elliptic estimates to obtain stability of Q under C^0 perturbations. A partial result on preservation of scalar curvature lower bounds under convergence in measure is also given.

Significance. If the main estimate holds with the stated dependence of constants solely on g0, this constitutes a significant advance by providing the first explicit quantitative control on scalar curvature under C^0 convergence in 3D, together with sharpness examples that clarify the optimal rate. The harmonic-function construction of the detecting quantity Q offers a flexible new technique that may extend to other questions involving curvature under weak topologies, as illustrated by the measure-convergence application.

major comments (1)

[Definition of the local quantity Q_g and the subsequent application of elliptic estimates] The central stability estimate |Q_g(0) - Q_{g0}(0)| ≤ C ||g-g0||_{C^0}^{1/2} (with C, ε0 depending only on g0) is obtained by applying classical Schauder or comparison estimates to the difference of harmonic functions for Δ_g and Δ_{g0}. These estimates produce constants that depend on the C^{1,α} or C^2 norms of the metric coefficients. The hypothesis ||g-g0||_{C^0} ≤ ε0 alone does not control these higher norms of g (high-frequency oscillations of amplitude ≤ ε0 can make ||∇^2 g|| arbitrarily large while keeping g smooth). It is therefore necessary to verify explicitly that the claimed dependence on g0 only is achieved; otherwise the bound on inf R_g cannot be guaranteed.

minor comments (1)

[Statement of Theorem 1.1 and the C^2 variant] Clarify the precise regularity assumed on g in the main theorem versus the C^2 variant; the abstract states both g and g0 are smooth but later refers to g0 merely C^2.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report, which highlights an important technical point in the stability argument for the quantity Q_g. We address the major comment below and will incorporate the necessary clarifications and verifications into a revised manuscript.

read point-by-point responses

Referee: The central stability estimate |Q_g(0) - Q_{g0}(0)| ≤ C ||g-g0||_{C^0}^{1/2} (with C, ε0 depending only on g0) is obtained by applying classical Schauder or comparison estimates to the difference of harmonic functions for Δ_g and Δ_{g0}. These estimates produce constants that depend on the C^{1,α} or C^2 norms of the metric coefficients. The hypothesis ||g-g0||_{C^0} ≤ ε0 alone does not control these higher norms of g (high-frequency oscillations of amplitude ≤ ε0 can make ||∇^2 g|| arbitrarily large while keeping g smooth). It is therefore necessary to verify explicitly that the claimed dependence on g0 only is achieved; otherwise the bound on inf R_g cannot be guaranteed.

Authors: We agree with the referee that the constants appearing in classical elliptic estimates must be tracked with care to ensure they depend only on g0. In the current proof, the stability of Q_g is derived from comparison principles applied to the difference of solutions to the Laplace-Beltrami equations with fixed boundary data. While the principal coefficients (the inverse metric components) are controlled by the C^0 closeness, the lower-order coefficients involve first derivatives of the metric and are not automatically bounded. We acknowledge that this dependence was not made fully explicit. In the revised manuscript we will insert a dedicated lemma that derives the required a priori bounds on all coefficients using only the uniform ellipticity constants (which depend on g0 and ε0) together with the maximum principle and Harnack-type inequalities whose constants are likewise controlled by these quantities. If this verification proves insufficient, we will instead replace the direct comparison argument by an integral-identity approach that avoids explicit derivative bounds on g. Either way, the claimed dependence of C and ε0 solely on g0 will be justified or the statement will be adjusted. revision: yes

Circularity Check

0 steps flagged

No circularity: independent construction and external elliptic theory

full rationale

The derivation introduces a novel local quantity Q_g constructed from g-harmonic functions (with g-independent boundary data) whose value at the origin is shown to detect R_g via direct computation from the metric and the harmonic equation. Stability |Q_g(0) - Q_{g0}(0)| ≤ C ||g - g0||_{C^0}^{1/2} is then obtained by applying standard, externally known elliptic estimates (Schauder or L^p theory) whose constants are asserted to depend only on g0. Neither the detection identity nor the stability bound is obtained by fitting parameters to the target inequality, by renaming a known result, or by any self-citation chain; the steps remain logically independent of the final scalar-curvature inequality. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The result rests on the existence of constants C and epsilon_0 supplied by elliptic theory once a curvature-detecting quantity is defined via harmonic functions; no new entities are postulated.

free parameters (2)

C
Positive constant whose existence is asserted to depend only on g0; not numerically fitted in the statement.
epsilon_0
Positive threshold whose existence is asserted to depend only on g0; not numerically fitted.

axioms (2)

standard math Classical elliptic regularity estimates hold for the Laplace-Beltrami operator associated to a C0-close metric
Invoked to control the stability of the harmonic-function quantity under C0 perturbations of the metric.
domain assumption Harmonic functions exist and can be used to define a local scalar-curvature detector on the ball
Central to the proof strategy described in the abstract.

pith-pipeline@v0.9.0 · 5569 in / 1539 out tokens · 32503 ms · 2026-05-10T11:45:41.029534+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Scalar curvature under weak limits of manifolds
math.DG 2026-05 unverdicted novelty 7.0

Scalar curvature lower bounds are preserved under weak limits of smooth closed 3-manifolds through μ-bubble comparisons when volumes and Lipschitz constants converge appropriately.
Quantification of scalar curvature under $C^0$ convergence using smoothing
math.DG 2026-04 unverdicted novelty 7.0

The refined quantitative scalar curvature lower bound under C^0 convergence holds in all dimensions greater than or equal to three.

Reference graph

Works this paper leans on

37 extracted references · 7 canonical work pages · cited by 2 Pith papers

[1]

A Green’s function proof of the positive mass theorem.Communications in Mathematical Physics, 405(2):54, 2024

Virginia Agostiniani, Lorenzo Mazzieri, and Francesca Oronzio. A Green’s function proof of the positive mass theorem.Communications in Mathematical Physics, 405(2):54, 2024

2024
[2]

On the stability of Llarull’s theorem in dimension three

Brian Allen, Edward Bryden, and Demetre Kazaras. On the stability of Llarull’s theorem in dimension three. Mathematische Annalen, 392(1):373–398, 2025

2025
[3]

A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature.Mathematical Research Letters, 23(2):325–337, 2016

Richard Bamler. A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature.Mathematical Research Letters, 23(2):325–337, 2016

2016
[4]

Rigidity results for initial data sets satisfying the dominant energy condition.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2025

Christian B¨ ar, Simon Brendle, Tsz-Kiu Aaron Chow, and Bernhard Hanke. Rigidity results for initial data sets satisfying the dominant energy condition.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2025

2025
[5]

Proof of the Riemannian Penrose inequality using the positive mass theorem.Journal of Differential Geometry, 59(2):177–267, 2001

Hubert L Bray. Proof of the Riemannian Penrose inequality using the positive mass theorem.Journal of Differential Geometry, 59(2):177–267, 2001

2001
[6]

Harmonic functions and the mass of 3-dimensional asymptotically flat Riemannian manifolds.The Journal of Geometric Analysis, 32(6):184, 2022

Hubert L Bray, Demetre P Kazaras, Marcus A Khuri, and Daniel L Stern. Harmonic functions and the mass of 3-dimensional asymptotically flat Riemannian manifolds.The Journal of Geometric Analysis, 32(6):184, 2022

2022
[7]

Scalar curvature rigidity of convex polytopes.Inventiones mathematicae, 235(2):669–708, 2024

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

2024
[8]

On Gromov’s rigidity theorem for polytopes with acute angles.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2025(827):191–225, 2025

Simon Brendle and Yipeng Wang. On Gromov’s rigidity theorem for polytopes with acute angles.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2025(827):191–225, 2025

2025
[9]

Pointwise lower scalar curvature bounds forC 0 metrics via regularizing Ricci flow.Geo- metric and Functional Analysis, 29(6):1703–1772, 2019

Paula Burkhardt-Guim. Pointwise lower scalar curvature bounds forC 0 metrics via regularizing Ricci flow.Geo- metric and Functional Analysis, 29(6):1703–1772, 2019

2019
[10]

Stability for the 3d Riemannian Penrose inequality.Geometry and Topology, 29(9):4911–4945, December 2025

Conghan Dong. Stability for the 3d Riemannian Penrose inequality.Geometry and Topology, 29(9):4911–4945, December 2025

2025
[11]

Stability of Euclidean 3-space for the positive mass theorem.Inventiones math- ematicae, 239(1):287–319, 2025

Conghan Dong and Antoine Song. Stability of Euclidean 3-space for the positive mass theorem.Inventiones math- ematicae, 239(1):287–319, 2025

2025
[12]

Dirac and Plateau billiards in domains with corners.Central European Journal of Mathematics, 12(8):1109–1156, 2014

Misha Gromov. Dirac and Plateau billiards in domains with corners.Central European Journal of Mathematics, 12(8):1109–1156, 2014

2014
[13]

Four lectures on scalar curvature.arXiv preprint arXiv:1908.10612, 2019

Misha Gromov. Four lectures on scalar curvature.arXiv preprint arXiv:1908.10612, 2019

work page arXiv 1908
[14]

Spin and scalar curvature in the presence of a fundamental group

Misha Gromov and H Blaine Lawson Jr. Spin and scalar curvature in the presence of a fundamental group. I. Annals of Mathematics, pages 209–230, 1980

1980
[15]

The Green function for uniformly elliptic equations.Manuscripta mathe- matica, 37(3):303–342, 1982

Michael Gr¨ uter and Kjell-Ove Widman. The Green function for uniformly elliptic equations.Manuscripta mathe- matica, 37(3):303–342, 1982

1982
[16]

The inverse mean curvature flow and the Riemannian Penrose inequality

Gerhard Huisken and Tom Ilmanen. The inverse mean curvature flow and the Riemannian Penrose inequality. Journal of Differential Geometry, 59(3):353–437, 2001. 32 LIAM MAZUROWSKI AND XUAN YAO

2001
[17]

Capillary minimal slicing and scalar curvature rigidity.arXiv preprint arXiv:2602.21071, 2026

Dongyeong Ko and Xuan Yao. Capillary minimal slicing and scalar curvature rigidity.arXiv preprint arXiv:2602.21071, 2026

work page arXiv 2026
[18]

Geometric flows with rough initial data.Asian Journal of Mathematics, 16(2):209– 235, 2012

Herbert Koch and Tobias Lamm. Geometric flows with rough initial data.Asian Journal of Mathematics, 16(2):209– 235, 2012

2012
[19]

Green functions and a positive mass theorem for asymptoti- cally hyperbolic 3-manifolds.arXiv preprint arXiv:2506.07108, 2025

Klaus Kroencke, Francesca Oronzio, and Alan Pinoy. Green functions and a positive mass theorem for asymptoti- cally hyperbolic 3-manifolds.arXiv preprint arXiv:2506.07108, 2025

work page arXiv 2025
[20]

A polyhedron comparison theorem for 3-manifolds with positive scalar curvature.Inventiones mathemat- icae, 219(1):1–37, 2020

Chao Li. A polyhedron comparison theorem for 3-manifolds with positive scalar curvature.Inventiones mathemat- icae, 219(1):1–37, 2020

2020
[21]

The dihedral rigidity conjecture forn-prisms.Journal of Differential Geometry, 126(1):329–361, 2024

Chao Li. The dihedral rigidity conjecture forn-prisms.Journal of Differential Geometry, 126(1):329–361, 2024

2024
[22]

Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries.Commu- nications on Pure and Applied Mathematics, 50(5):449–487, 1997

Yanyan Li and Meijun Zhu. Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries.Commu- nications on Pure and Applied Mathematics, 50(5):449–487, 1997

1997
[23]

Sharp estimates and the Dirac operator.Mathematische Annalen, 310(1):55–71, 1998

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

1998
[24]

Rigidity of min-max minimal spheres in three-manifolds.Duke Mathematical Journal, 161(14):2725, 2012

Fernando C Marques and Andr´ e Neves. Rigidity of min-max minimal spheres in three-manifolds.Duke Mathematical Journal, 161(14):2725, 2012

2012
[25]

The Yamabe invariant ofRP 3 via harmonic functions.arXiv preprint arXiv:2305.00854, 2023

Liam Mazurowski and Xuan Yao. The Yamabe invariant ofRP 3 via harmonic functions.arXiv preprint arXiv:2305.00854, 2023

work page arXiv 2023
[26]

On the stability of the Yamabe invariant ofS 3.arXiv preprint arXiv:2402.00815, 2024

Liam Mazurowski and Xuan Yao. On the stability of the Yamabe invariant ofS 3.arXiv preprint arXiv:2402.00815, 2024

work page arXiv 2024
[27]

Monotone quantities forp-harmonic functions and the sharpp-Penrose inequality

Liam Mazurowski and Xuan Yao. Monotone quantities forp-harmonic functions and the sharpp-Penrose inequality. Mathematical Research Letters, 32(3):957–995, 2025

2025
[28]

AnL p-estimate for the gradient of solutions of second order elliptic divergence equations

Norman G Meyers. AnL p-estimate for the gradient of solutions of second order elliptic divergence equations. Annali della Scuola Normale Superiore di Pisa-Scienze Fisiche e Matematiche, 17(3):189–206, 1963

1963
[29]

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

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

2025
[30]

Comparison theorems for three-dimensional manifolds with scalar curvature bound.arXiv preprint arXiv:2105.12103, 2021

Ovidiu Munteanu and Jiaping Wang. Comparison theorems for three-dimensional manifolds with scalar curvature bound.arXiv preprint arXiv:2105.12103, 2021

work page arXiv 2021
[31]

Springer, 2006

Peter Petersen.Riemannian geometry. Springer, 2006

2006
[32]

Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature.Annals of Mathematics, 110(1):127–142, 1979

Richard Schoen and Shing-Tung Yau. Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature.Annals of Mathematics, 110(1):127–142, 1979

1979
[33]

On the proof of the positive mass conjecture in general relativity.Commu- nications in Mathematical Physics, 65(1):45–76, 1979

Richard Schoen and Shing-Tung Yau. On the proof of the positive mass conjecture in general relativity.Commu- nications in Mathematical Physics, 65(1):45–76, 1979

1979
[34]

Scalar curvature and harmonic maps toS 1.Journal of Differential Geometry, 122(2):259–269, 2022

Daniel L Stern. Scalar curvature and harmonic maps toS 1.Journal of Differential Geometry, 122(2):259–269, 2022

2022
[35]

On gromov’s dihedral extremality and rigidity conjectures.arXiv preprint arXiv:2112.01510, 2021

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

work page arXiv 2021
[36]

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

2024
[37]

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. In Annales Henri Poincar´ e, volume 17, pages 2265–2283. Springer, 2016. Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania, 18015, United States Email address:lim624@lehigh.edu Department of Mathematics, Princeton University, Prin...

2016