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arxiv: 2606.23272 · v1 · pith:Z5L3PFKUnew · submitted 2026-06-22 · 🧮 math.DG · math.AP

Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass

Man-Chun Lee , Florian Litzinger , Miles Simon This is my paper

Pith reviewed 2026-06-26 07:05 UTC · model grok-4.3

classification 🧮 math.DG math.AP
keywords positive mass theoremADM massdistributional scalar curvatureRicci flowasymptotically flat manifoldslow regularityoptimal regularityintegral distance
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The pith

Asymptotically flat manifolds of regularity L^∞ ∩ W^{1,n} with non-negative distributional scalar curvature have non-negative ADM mass.

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

The paper establishes that the positive mass theorem holds for asymptotically flat manifolds whose metrics lie in the critical regularity class L^∞ ∩ W^{1,n} and whose scalar curvature is non-negative in the distributional sense. It proves that the ADM mass is non-negative, and that vanishing ADM mass forces the manifold to be globally isometric to Euclidean space with respect to a specific integral distance. The argument proceeds by constructing a Ricci flow smoothing that turns the distributional lower bound into a classical one. A reader would care because the result identifies the precise regularity threshold at which the theorem remains valid, in contrast to strictly lower integrability where counterexamples exist.

Core claim

We show that asymptotically flat manifolds (M^n,g) of regularity L^∞∩W^{1,n} with non-negative distributional scalar curvature have non-negative ADM mass. Furthermore, when the ADM mass vanishes, the manifold is globally isometric to Euclidean space with respect to an integral distance introduced by De Cecco-Palmieri. Our approach is based on showing that Riemannian metrics of regularity L^∞∩W^{1,n}, whose scalar curvature is bounded from below in the distributional sense, admit a Ricci flow smoothing whose scalar curvature is bounded from below by the same initial lower bound in the classical sense.

What carries the argument

Ricci flow smoothing that converts a distributional scalar curvature lower bound on an L^∞ ∩ W^{1,n} metric into a classical lower bound.

If this is right

  • The ADM mass is non-negative for all such manifolds.
  • Vanishing ADM mass implies global isometry to Euclidean space in the De Cecco-Palmieri integral distance.
  • The regularity L^∞ ∩ W^{1,n} is optimal, since for every p < n there exist L^∞ ∩ W^{1,p} metrics with distributional scalar curvature bounded below that cannot be approximated by smooth metrics preserving the bound.
  • The positive mass theorem extends to this critical regularity via the smoothing construction.

Where Pith is reading between the lines

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

  • The smoothing method may extend to other geometric inequalities that rely on scalar curvature lower bounds in low regularity.
  • It raises the question whether the integral-distance flatness implies pointwise flatness or other rigidity properties under the same assumptions.
  • Similar Ricci-flow approximations could be tested on non-asymptotically-flat manifolds with distributional curvature bounds.

Load-bearing premise

Metrics of regularity L^∞ ∩ W^{1,n} whose scalar curvature is bounded below distributionally admit a Ricci flow smoothing that preserves the same lower bound in the classical sense.

What would settle it

An asymptotically flat manifold carrying an L^∞ ∩ W^{1,n} metric with non-negative distributional scalar curvature yet negative ADM mass would falsify the claim.

read the original abstract

In this work, we study the positive mass theorem under critical low regularity assumptions using Ricci flow smoothing. We show that asymptotically flat manifolds $(M^n,g)$ of regularity $L^\infty\cap W^{1,n}$ with non-negative distributional scalar curvature have non-negative ADM mass. Furthermore, when the ADM mass vanishes, the manifold is globally isometric to Euclidean space with respect to an integral distance introduced by De~Cecco-Palmieri. This extends the recent work of Hafemann to the critical regularity case. Our approach is based on showing that Riemannian metrics of regularity $L^\infty\cap W^{1,n}$, whose scalar curvature is bounded from below in the distributional sense, admit a Ricci flow smoothing whose scalar curvature is bounded from below by the same initial lower bound in the classical sense. In contrast, Cecchini-Frenck-Zeidler constructed examples of metrics which are in $L^\infty\cap W^{1,p}$ for all $2<p<n$, and whose distributional scalar curvature is bounded from below, that cannot be approximated by smooth metrics with the same scalar curvature lower bound. In this sense, our result is optimal.

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.

Referee Report

0 major / 2 minor

Summary. The paper claims that asymptotically flat manifolds (M^n, g) of regularity L^∞ ∩ W^{1,n} with non-negative distributional scalar curvature have non-negative ADM mass. When the ADM mass vanishes, the manifold is globally isometric to Euclidean space with respect to the De Cecco-Palmieri integral distance. The proof proceeds by constructing a Ricci flow smoothing of such metrics that preserves the lower bound on scalar curvature in the classical sense, then applying the smooth positive mass theorem and passing to the limit. This is presented as optimal relative to the Cecchini–Frenck–Zeidler counterexamples for p < n, and extends Hafemann's prior work to the critical regularity case.

Significance. If the central construction holds, the result is significant: it establishes the positive mass theorem at the critical regularity L^∞ ∩ W^{1,n} for distributional scalar curvature bounds, achieving optimality in light of the cited counterexamples. The Ricci flow smoothing approach, if the error estimates and preservation of the curvature bound are rigorous, provides a clean reduction to the smooth case and yields a rigidity statement via the integral distance.

minor comments (2)
  1. The abstract and introduction should explicitly reference the section containing the Ricci flow smoothing construction (including the precise statement of the preservation of the scalar curvature lower bound) so that readers can locate the load-bearing technical step.
  2. Clarify the precise sense in which the limit of the smoothed metrics recovers the original ADM mass; a short paragraph or remark after the main theorem would help.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our main results on the positive mass theorem at critical regularity L^∞ ∩ W^{1,n} via Ricci flow smoothing, and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation applies external Ricci flow and positive mass results to new regularity class

full rationale

The paper's central argument constructs a Ricci flow smoothing for L^∞ ∩ W^{1,n} metrics that preserves the distributional scalar curvature lower bound in the classical sense, then invokes the smooth positive mass theorem and passes to the limit. This chain relies on established external techniques (Ricci flow smoothing, smooth PMT) and contrasts with counterexamples by Cecchini–Frenck–Zeidler; the extension of Hafemann's work is cited as prior independent input rather than a self-citation chain. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract: the proof relies on the existence of a Ricci flow smoothing that preserves the distributional lower bound on scalar curvature for the given regularity class. No free parameters, invented entities, or ad-hoc axioms are mentioned.

axioms (1)
  • domain assumption Existence of Ricci flow smoothing for L^∞ ∩ W^{1,n} metrics that preserves the distributional scalar curvature lower bound in the classical sense after smoothing.
    This is the key technical step described in the abstract as the basis for applying the smooth positive mass theorem.

pith-pipeline@v0.9.1-grok · 5731 in / 1273 out tokens · 26032 ms · 2026-06-26T07:05:03.527915+00:00 · methodology

discussion (0)

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