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arxiv: 2606.15372 · v2 · pith:RQJSEGOAnew · submitted 2026-06-13 · 🧮 math.DG

Gromov's Euclidean Endpoint C⁰ Rigidity for the Positive Mass Theorem

Jiangcheng You , Heng Zhang This is my paper

Pith reviewed 2026-06-27 04:11 UTC · model grok-4.3

classification 🧮 math.DG
keywords positive mass theoremscalar curvaturerigidityasymptotically flateuclidean metricC0 rigidityGromov conjecture
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The pith

A smooth complete metric on R^3 with nonnegative scalar curvature and o(r^{-1}) decay at infinity is isometric to Euclidean space.

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

The paper proves Gromov's conjecture on the Euclidean endpoint C^0 rigidity. It shows that nonnegative scalar curvature everywhere on R^3, together with the metric approaching the flat metric faster than order 1/r at infinity, forces the metric to be exactly Euclidean. This matters because it identifies the precise decay threshold at which the positive mass theorem implies global flatness rather than just zero mass at infinity. The result closes the gap between known rigidity statements and the weakest possible asymptotic condition in three dimensions.

Core claim

Let g be a smooth complete metric on R^3 with nonnegative scalar curvature. If |g - g_Euc| = o(r^{-1}) as r = |x| tends to infinity, then (R^3, g) is isometric to Euclidean space.

What carries the argument

The o(r^{-1}) decay condition at infinity combined with the global nonnegativity of scalar curvature, which together trigger rigidity conclusions from the positive mass theorem.

If this is right

  • The mass of any such manifold must vanish.
  • The manifold is flat at every point, not merely asymptotically.
  • The positive mass theorem remains valid at this endpoint decay rate.
  • No non-flat example exists under these exact hypotheses.

Where Pith is reading between the lines

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

  • The same decay and curvature conditions may classify all possible isolated systems in this asymptotic class.
  • Slower decay rates could permit curvature if the nonnegativity assumption is relaxed at finite distances.
  • Numerical constructions attempting to approximate such metrics should converge to flat space or violate one hypothesis.

Load-bearing premise

The o(r^{-1}) decay at infinity is strong enough, when scalar curvature stays nonnegative everywhere, to force the metric to be exactly flat via geometric analysis.

What would settle it

A counterexample would be any non-flat complete metric g on R^3 with scalar curvature nonnegative at every point yet satisfying |g minus the Euclidean metric| = o(r^{-1}) at large r.

read the original abstract

We prove Gromov's Euclidean endpoint $C^0$ rigidity conjecture. Let $g$ be a smooth complete metric on $\R^3$ with non-negative scalar curvature. If $$ |g-g_{\Euc}|=o(r^{-1}),\qquad r=|x|\to\infty, $$ then $(\R^3,g)$ is isometric to Euclidean space.

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

2 major / 0 minor

Summary. The paper claims to prove Gromov's Euclidean endpoint C^0 rigidity conjecture for the positive mass theorem: Let g be a smooth complete metric on R^3 with non-negative scalar curvature. If |g - g_Euc| = o(r^{-1}) as r = |x| -> infinity, then (R^3, g) is isometric to Euclidean space.

Significance. If the result holds, it would strengthen the rigidity statement associated to the positive mass theorem by relaxing the standard O(r^{-1}) decay to the weaker little-o condition, potentially clarifying the precise asymptotic requirements for Euclidean rigidity in geometric analysis.

major comments (2)
  1. [Abstract] Abstract (theorem statement): the reduction to known PMT rigidity requires that the ADM mass be well-defined and vanish under the given decay. The o(r^{-1}) condition alone does not guarantee convergence of the surface integrals at infinity (standard definitions require O(r^{-1}) decay together with O(r^{-2}) on derivatives), so an auxiliary argument establishing mass vanishing must be supplied; without it the central claim does not follow from existing rigidity theorems.
  2. [§2 or §3] The asymptotic analysis (likely §2 or §3): the manuscript must explicitly verify that the little-o decay implies the mass integrand yields a finite limit that is necessarily zero when scalar curvature is non-negative, rather than assuming this step follows automatically from the statement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight an important point regarding the well-definedness of the ADM mass under the little-o decay. We address each major comment below and will revise the manuscript accordingly to make the asymptotic analysis fully explicit.

read point-by-point responses

  1. Referee: [Abstract] Abstract (theorem statement): the reduction to known PMT rigidity requires that the ADM mass be well-defined and vanish under the given decay. The o(r^{-1}) condition alone does not guarantee convergence of the surface integrals at infinity (standard definitions require O(r^{-1}) decay together with O(r^{-2}) on derivatives), so an auxiliary argument establishing mass vanishing must be supplied; without it the central claim does not follow from existing rigidity theorems.

    Authors: We agree that the little-o decay does not automatically guarantee the standard hypotheses for the ADM mass. The manuscript's proof strategy relies on reducing to known rigidity results once mass vanishing is established, but the referee is correct that an auxiliary argument is required to justify that the surface integrals converge to a limit of zero. We will add this verification explicitly in a new subsection of §2, using the non-negative scalar curvature to control the decay of the error terms in the integrand and show that the mass must vanish. revision: yes

  2. Referee: [§2 or §3] The asymptotic analysis (likely §2 or §3): the manuscript must explicitly verify that the little-o decay implies the mass integrand yields a finite limit that is necessarily zero when scalar curvature is non-negative, rather than assuming this step follows automatically from the statement.

    Authors: We accept this criticism. The current draft treats the passage from little-o decay plus Scal ≥ 0 to mass vanishing as following from standard estimates, but it should be spelled out. The revision will include a self-contained lemma proving that the integrand has a well-defined limit equal to zero under the stated hypotheses, thereby closing the reduction to the known PMT rigidity theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct proof of stated conjecture

full rationale

The paper states and claims to prove the theorem directly from non-negative scalar curvature plus the given little-o decay on R^3. No quoted equations or sections exhibit self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citation chains that collapse the central claim to its inputs by construction. The derivation is presented as self-contained geometric analysis, consistent with external benchmarks such as the positive mass theorem rigidity statements under the stated hypotheses. This is the normal, non-circular outcome for a direct proof manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard framework of Riemannian geometry (smooth manifolds, scalar curvature, completeness) and the positive mass theorem literature. No free parameters or invented entities are visible in the abstract. Specific axioms invoked inside the proof are not accessible from the abstract alone.

axioms (1)
  • standard math Standard axioms of smooth Riemannian geometry on complete manifolds
    The setting is a smooth complete metric on R^3.

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Forward citations

Cited by 1 Pith paper

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

  1. A Positive Mass Theorem for Continuous Metrics

    math.DG 2026-06 unverdicted novelty 7.0

    Proves that the harmonic mass of a continuous asymptotically flat metric on R^3 is non-negative, with equality only when the metric is flat.

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