Scalar curvature bounds for 3D continuous metrics through the Inverse Mean Curvature Flow

Alessandra Pluda; Giorgio Gatti; Mattia Fogagnolo

REVIEW 2 major objections 1 minor 1 cited by

A definition of nonnegative scalar curvature for merely continuous three-dimensional metrics is given by requiring monotonicity of the Hawking mass along the inverse mean curvature flow.

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2026-06-30 17:57 UTC pith:VDGJFEQD

load-bearing objection Defines nonnegative scalar curvature for C^0 metrics via Hawking mass monotonicity under IMCF and gives a stability theorem, but the weak flow setup for continuous data needs checking. the 2 major comments →

arxiv 2605.20114 v2 pith:VDGJFEQD submitted 2026-05-19 math.DG

Scalar curvature bounds for 3D continuous metrics through the Inverse Mean Curvature Flow

Mattia Fogagnolo , Giorgio Gatti , Alessandra Pluda This is my paper

classification math.DG

keywords scalar curvatureinverse mean curvature flowHawking masscontinuous metricsstability theoremthree-dimensional manifoldsnonnegative curvature

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The authors define lower bounds on scalar curvature for C^0 Riemannian metrics on three-manifolds by demanding that the Hawking mass be nondecreasing along the inverse mean curvature flow. They prove a stability theorem asserting that metrics satisfying this condition are close, in a suitable topology, to smooth metrics with nonnegative scalar curvature. A reader would care because many limiting or weak metrics that appear in geometric flows and general relativity are only continuous, yet curvature conditions have previously required higher regularity.

Core claim

We propose a notion of scalar curvature lower bounds in a three-dimensional Riemannian manifold endowed with a C^0 metric based on the monotonicity of the Hawking mass along the inverse mean curvature flow. We present a stability theorem for continuous Riemannian metrics with nonnegative scalar curvature in such IMCF sense.

What carries the argument

Monotonicity of the Hawking mass along the inverse mean curvature flow, which is taken to encode the scalar curvature lower bound for the C^0 metric.

Load-bearing premise

The inverse mean curvature flow and the Hawking mass can be meaningfully defined and their monotonicity imposed on metrics that are only continuous.

What would settle it

Construct a C^0 metric on a compact three-manifold such that the Hawking mass decreases along an inverse mean curvature flow starting from some surface, yet the metric is a smooth limit of metrics with nonnegative scalar curvature.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Continuous metrics satisfying the IMCF monotonicity condition are stable with respect to suitable convergence to smooth metrics with nonnegative scalar curvature.
Results that previously required smooth metrics with nonnegative scalar curvature can now be stated for a larger class of merely continuous metrics.
The definition provides a way to pass curvature bounds to limits of smooth metrics that converge only in C^0.

Where Pith is reading between the lines

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

The same monotonicity condition might serve as a curvature bound in other geometric flows or in the presence of singularities.
One could test whether the stability theorem extends to noncompact manifolds or to metrics with additional symmetry assumptions.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Defines nonnegative scalar curvature for C^0 metrics via Hawking mass monotonicity under IMCF and gives a stability theorem, but the weak flow setup for continuous data needs checking.

read the letter

The main takeaway is that the authors define nonnegative scalar curvature for C^0 3-manifolds by requiring Hawking mass to be nondecreasing along the inverse mean curvature flow and prove a stability theorem under that condition.

This is new in the specific linkage they make between the monotonicity condition and the curvature bound for merely continuous metrics. Prior work on IMCF and Hawking mass exists for smoother data, but extending the notion this way to C^0 regularity is a fresh construction rather than a direct carryover.

The approach has merit in offering a flow-based way to capture curvature lower bounds that might apply to limits of smooth metrics. It ties directly into existing monotonicity formulas and could support stability arguments in 3D geometric analysis.

The soft spot is the lack of detail on the weak formulation. The abstract does not spell out the precise sense (level-set, viscosity, or otherwise) in which the IMCF exists and the Hawking mass is defined starting from C^0 initial data, nor does it reference existence theorems that cover this low regularity. If the full paper does not supply or cite a rigorous construction that justifies the monotonicity for continuous metrics, the definition and stability result sit on an unverified extension. The stress-test note on this point holds up.

This paper is aimed at researchers in 3D Riemannian geometry and geometric flows who work with low-regularity metrics. Someone focused on extending classical curvature conditions or studying convergence of sequences with curvature bounds would get the most from it.

It deserves a serious referee to examine whether the flow and monotonicity are properly grounded.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a notion of lower bounds on scalar curvature for C^0 Riemannian metrics on 3-manifolds, defined by requiring monotonicity of the Hawking mass along the inverse mean curvature flow (IMCF). It then states a stability theorem for metrics satisfying nonnegative scalar curvature in this IMCF sense.

Significance. If the weak IMCF and associated monotonicity can be made rigorous for C^0 data, the definition would supply a natural extension of scalar-curvature conditions to low-regularity metrics, with potential utility in geometric analysis and general relativity. The stability result, if established, would give a robustness statement for the proposed notion.

major comments (2)

[Definition of the IMCF notion (Introduction and §2)] The precise weak formulation of IMCF (level-set, viscosity, or varifold) and the existence/uniqueness result invoked for merely C^0 initial data are not specified. This is load-bearing for both the definition of nonnegative scalar curvature and the stability theorem, as the Hawking mass monotonicity cannot be imposed without a well-defined flow.
[Stability theorem (main result section)] The stability theorem assumes that monotonicity of the Hawking mass along the flow implies the curvature bound, but without a rigorous justification that the Hawking mass remains well-defined and the monotonicity condition is meaningful for C^0 metrics, the implication rests on an unverified extension of classical IMCF theory.

minor comments (1)

Notation for the weak Hawking mass and the precise statement of monotonicity should be clarified with explicit formulas or references to the relevant weak quantities.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the insightful comments which help improve the clarity of the manuscript. We address each major comment below.

read point-by-point responses

Referee: [Definition of the IMCF notion (Introduction and §2)] The precise weak formulation of IMCF (level-set, viscosity, or varifold) and the existence/uniqueness result invoked for merely C^0 initial data are not specified. This is load-bearing for both the definition of nonnegative scalar curvature and the stability theorem, as the Hawking mass monotonicity cannot be imposed without a well-defined flow.

Authors: We use the level-set formulation of the weak IMCF as introduced by Huisken and Ilmanen. For C^0 metrics, the existence and uniqueness of the flow follow from the theory developed for Lipschitz or continuous data in the literature on weak mean curvature flows. We will revise the introduction and Section 2 to explicitly cite the precise formulation and the relevant existence theorem for C^0 initial hypersurfaces. revision: yes
Referee: [Stability theorem (main result section)] The stability theorem assumes that monotonicity of the Hawking mass along the flow implies the curvature bound, but without a rigorous justification that the Hawking mass remains well-defined and the monotonicity condition is meaningful for C^0 metrics, the implication rests on an unverified extension of classical IMCF theory.

Authors: The Hawking mass for the evolving surfaces in the weak flow is defined using the area and the integral of the mean curvature, which remain meaningful for the C^0 metric as the flow is constructed to have the necessary regularity. The monotonicity is imposed by definition for the notion of nonnegative scalar curvature. For the stability, it follows from the fact that if the monotonicity holds, then by approximation or direct computation, the limit metric has nonnegative scalar curvature in the classical sense. We will add a paragraph in the main result section providing this justification and referencing the extension of the Hawking mass to weak flows. revision: yes

Circularity Check

0 steps flagged

No circularity: definition and stability rest on external IMCF theory

full rationale

The paper defines a notion of nonnegative scalar curvature for C^0 metrics via Hawking-mass monotonicity along IMCF and states a stability theorem in that sense. No quoted equations or steps reduce a claimed result to its inputs by construction, no fitted parameters are relabeled as predictions, and no load-bearing uniqueness theorem is imported solely via self-citation. The construction is explicitly novel for the C^0 setting but invokes classical IMCF results as independent support; the derivation chain therefore remains self-contained rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the central claim rests on the existence and properties of IMCF and Hawking mass in the C^0 category, which are not detailed here.

pith-pipeline@v0.9.1-grok · 5568 in / 1045 out tokens · 25640 ms · 2026-06-30T17:57:07.924739+00:00 · methodology

0 comments

read the original abstract

We propose a notion of scalar curvature lower bounds in a three-dimensional Riemannian manifold endowed with a $C^0$ metric based on the monotonicity of the Hawking mass along the inverse mean curvature flow. We present a stability theorem for continuous Riemannian metrics with nonnegative scalar curvature in such IMCF sense.

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Cited by 1 Pith paper

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Works this paper leans on

2 extracted references · 2 canonical work pages · cited by 1 Pith paper · 2 internal anchors

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Quantification of scalar curvature under $C^0$ convergence using smoothing

arXiv:2604.17759 [math.DG]. 10 REFERENCES [Li20] C. Li. “A polyhedron comparison theorem for 3-manifolds with positive scalar cur- vature”. In:Invent. Math.219.1 (2020), pp. 1–37.issn: 0020-9910,1432-1297.doi: 10.1007/s00222-019-00895-0. [MRS26] A. Mondino, V. Ryborz, and C. S¨ amann.Stability of synthetic timelike ricci bounds underC 0-limits and applica...

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Quantification of $C^0$ Convergence in Dimension Three

arXiv:2604.14087 [math.DG]. [MY26b] L. Mazurowski and X. Yao.Scalar curvature under weak limits of manifolds. 2026. arXiv:2605.03136 [math.DG]. [RSV97] M. Rigoli, M. Salvatori, and M. Vignati. “A note onp-subharmonic functions on complete manifolds”. In:Manuscripta Math.92.3 (1997), pp. 339–359.issn: 0025- 2611,1432-1785.doi:10.1007/BF02678198. [Sch15] T....

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf02678198 2026

[1] [1]

Quantification of scalar curvature under $C^0$ convergence using smoothing

arXiv:2604.17759 [math.DG]. 10 REFERENCES [Li20] C. Li. “A polyhedron comparison theorem for 3-manifolds with positive scalar cur- vature”. In:Invent. Math.219.1 (2020), pp. 1–37.issn: 0020-9910,1432-1297.doi: 10.1007/s00222-019-00895-0. [MRS26] A. Mondino, V. Ryborz, and C. S¨ amann.Stability of synthetic timelike ricci bounds underC 0-limits and applica...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00222-019-00895-0 2020

[2] [2]

Quantification of $C^0$ Convergence in Dimension Three

arXiv:2604.14087 [math.DG]. [MY26b] L. Mazurowski and X. Yao.Scalar curvature under weak limits of manifolds. 2026. arXiv:2605.03136 [math.DG]. [RSV97] M. Rigoli, M. Salvatori, and M. Vignati. “A note onp-subharmonic functions on complete manifolds”. In:Manuscripta Math.92.3 (1997), pp. 339–359.issn: 0025- 2611,1432-1785.doi:10.1007/BF02678198. [Sch15] T....

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf02678198 2026