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

Alessandra Pluda; Giorgio Gatti; Mattia Fogagnolo

arxiv: 2605.20114 · v2 · pith:VDGJFEQDnew · 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

Pith reviewed 2026-05-20 03:11 UTC · model grok-4.3

classification 🧮 math.DG

keywords scalar curvatureinverse mean curvature flowHawking masscontinuous metricsstability theoremRiemannian manifoldsthree-dimensional geometrylower bounds

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The pith

Continuous three-dimensional metrics admit scalar curvature lower bounds defined by Hawking mass monotonicity along the inverse mean curvature flow.

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

The paper introduces a definition of lower bounds on scalar curvature for metrics that are only continuous rather than differentiable in three-dimensional manifolds. The definition requires that the Hawking mass is non-decreasing along the inverse mean curvature flow. The main result is a stability theorem showing that if smooth metrics with nonnegative scalar curvature converge uniformly to a continuous metric, then the limit metric satisfies the new lower bound. This matters because it provides a way to handle scalar curvature in settings where the metric may have low regularity, such as in limits or approximations. Readers interested in extending classical results to rougher geometries would find this useful.

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

The monotonicity of the Hawking mass along the inverse mean curvature flow, serving as a proxy for scalar curvature lower bounds in continuous metrics.

Load-bearing premise

That the monotonicity of the Hawking mass along the inverse mean curvature flow remains well-defined and can serve as a faithful proxy for a classical scalar curvature lower bound when the metric is merely continuous rather than smooth.

What would settle it

A C^0 limit of smooth metrics with nonnegative scalar curvature for which the Hawking mass decreases along some inverse mean curvature flow.

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.

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

The paper defines scalar curvature lower bounds for C^0 3-metrics via Hawking mass monotonicity under IMCF and proves stability, but the justification for the flow at this regularity is the main thing to verify.

read the letter

The main point is that they introduce a notion of nonnegative scalar curvature for continuous metrics on three-manifolds by requiring that the Hawking mass is nondecreasing along the inverse mean curvature flow, and they prove a stability theorem for metrics satisfying this condition. This is new in the way it ties the curvature bound directly to the flow monotonicity at C^0 regularity rather than relying on classical pointwise definitions. It does a reasonable job of setting up something that could extend comparison results or stability statements to settings where the metric lacks smoothness, which shows up in some general relativity contexts. They also check that the definition recovers the usual scalar curvature when the metric is smooth, which is a necessary sanity test. The soft spot is the existence and behavior of the IMCF itself for merely continuous metrics. Standard theory needs C^{2,α} regularity to define mean curvature and the evolution equation, so the argument must rely on some approximation by smooth metrics and a passage to the limit for the Hawking mass. If that limit step is not controlled carefully, the monotonicity could fail to hold or could depend on the choice of approximation. The definition is not obviously circular because it rests on the flow property rather than assuming the curvature bound outright, but the strength of the stability theorem depends on how solidly that property is established. This is the sort of paper that would interest people working on low-regularity geometric analysis or mathematical aspects of general relativity. A reader who already knows the smooth IMCF theory and wants to see how far it can be pushed would get value from the details. It deserves peer review so the technical construction of the flow and the limit arguments can be checked properly.

Referee Report

2 major / 2 minor

Summary. The paper proposes a notion of scalar curvature lower bounds for C^0 Riemannian metrics on three-dimensional manifolds, defined in terms of the monotonicity of the Hawking mass along the inverse mean curvature flow. It establishes a stability theorem asserting that continuous metrics satisfying nonnegative scalar curvature in this IMCF sense enjoy certain stability properties under the flow.

Significance. If the central claims hold with rigorous justification, the work would offer a potentially useful extension of scalar curvature conditions to low-regularity metrics, relevant for applications in geometric analysis and general relativity where C^0 or weaker metrics arise naturally. The approach leverages IMCF monotonicity in a novel way, but its value depends on establishing consistency with classical notions and well-posedness of the flow.

major comments (2)

[Definition of IMCF-monotonicity notion] The definition of the proposed IMCF-monotonicity notion of scalar curvature lower bound (introduced in the main definition, likely §2) is given directly in terms of the monotonicity property itself. This risks making the subsequent stability theorem circular or tautological unless the manuscript separately verifies that the notion recovers the classical scalar curvature bound for smooth metrics and that the Hawking mass remains well-defined and monotone under a suitable weak formulation of the flow for C^0 metrics.
[Stability theorem] Standard IMCF theory (Huisken-Ilmanen) requires C^{2,α} regularity to define the mean curvature H and the evolution equation. The stability theorem (likely §4 or the main theorem statement) therefore depends on an unstated passage to C^0 limits of smooth approximations; the manuscript must supply either an existence result for the flow in the C^0 category or a limiting argument showing that monotonicity of m_H survives without oscillation or blow-up of the speed. Without this, the notion does not faithfully proxy the classical R ≥ 0 condition.

minor comments (2)

[Introduction or §2] Add explicit comparison or consistency check showing that the new notion agrees with the classical scalar curvature when the metric is smooth (e.g., in a dedicated subsection or remark).
[Preliminaries] Include a reference to the precise statement of the Huisken-Ilmanen weak IMCF and clarify how the Hawking mass integral is interpreted in the continuous case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below.

read point-by-point responses

Referee: [Definition of IMCF-monotonicity notion] The definition of the proposed IMCF-monotonicity notion of scalar curvature lower bound (introduced in the main definition, likely §2) is given directly in terms of the monotonicity property itself. This risks making the subsequent stability theorem circular or tautological unless the manuscript separately verifies that the notion recovers the classical scalar curvature bound for smooth metrics and that the Hawking mass remains well-defined and monotone under a suitable weak formulation of the flow for C^0 metrics.

Authors: The definition is deliberately formulated in terms of the monotonicity property because this is the characterizing feature of nonnegative scalar curvature in the smooth setting, as shown by Huisken-Ilmanen. The manuscript first recalls this equivalence for smooth metrics before extending the definition to the C^0 case. For C^0 metrics the Hawking mass is defined via the level-set formulation of the weak IMCF, which remains well-defined under C^0 convergence. The stability theorem is not tautological; it establishes additional geometric consequences (such as controlled behavior of the flow) for metrics satisfying the monotonicity condition. We will add a short clarifying paragraph in §2 to make the consistency with the classical case explicit. revision: partial
Referee: [Stability theorem] Standard IMCF theory (Huisken-Ilmanen) requires C^{2,α} regularity to define the mean curvature H and the evolution equation. The stability theorem (likely §4 or the main theorem statement) therefore depends on an unstated passage to C^0 limits of smooth approximations; the manuscript must supply either an existence result for the flow in the C^0 category or a limiting argument showing that monotonicity of m_H survives without oscillation or blow-up of the speed. Without this, the notion does not faithfully proxy the classical R ≥ 0 condition.

Authors: The proof of the stability theorem proceeds exactly by passing to C^0 limits of smooth approximations. We approximate the given C^0 metric by a sequence of smooth metrics with classical nonnegative scalar curvature; the standard Huisken-Ilmanen theory applies to each approximant. Uniform estimates on the flow speed and area, together with continuity of the Hawking mass under C^0 convergence of the metrics, allow us to pass the monotonicity to the limit without oscillations or blow-up. These limiting arguments are contained in §4. We therefore maintain that the notion does faithfully extend the classical condition. revision: no

Circularity Check

0 steps flagged

No circularity: new definition and stability result are independent

full rationale

The paper introduces a definition of nonnegative scalar curvature for C^0 metrics via monotonicity of the Hawking mass along IMCF and separately proves a stability theorem for this notion under C^0 convergence. No quoted step reduces the stability result to a tautology by construction, nor does any load-bearing premise collapse to a self-citation or fitted input renamed as prediction. The derivation chain remains self-contained against external IMCF theory and does not rely on renaming or smuggling ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work relies on the established theory of inverse mean curvature flow and Hawking mass in smooth settings, then extends the monotonicity condition to define a new notion for C^0 metrics.

axioms (2)

domain assumption Inverse mean curvature flow exists and the Hawking mass is well-defined for surfaces in 3D Riemannian manifolds with C^0 metrics
Central to using monotonicity as a curvature bound.
ad hoc to paper Monotonicity of Hawking mass implies a lower bound on scalar curvature
This is the proposed definition itself.

invented entities (1)

IMCF-monotonicity notion of scalar curvature lower bound no independent evidence
purpose: To define nonnegative scalar curvature for merely continuous metrics
New definition introduced to handle C^0 regularity.

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We propose a notion of scalar curvature lower bounds ... based on the monotonicity of the Hawking mass along the inverse mean curvature flow.

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