Rigidity in the Positive Mass Theorem with $C^0$ Decay

Abstract: Let $g$ be a smooth metric on $\mathbb R<sup>3$ with non-negative scalar curvature. We show that if $g$ satisfies $$\vert g(x)-g_{\text{euc}}(x)\vert = O(\vert x\vert<sup>{-1-τ})$$ for some $τ&gt; 0$ then $g$ must be flat.

• The paper establishes that smooth Riemannian metrics on ℝ³ with O(|x|⁻¹⁻τ) decay, τ>0, and nonnegative scalar curvature must be flat, thereby confirming rigidity.
• It leverages harmonic function techniques, monotonicity functionals, and elliptic regularity to derive sharp asymptotic expansions for the Green’s function.
• The work constructs counterexamples at the threshold decay O(|x|⁻¹), highlighting the precise boundary for rigidity in the positive mass theorem.

Rigidity in the Positive Mass Theorem with $C^0$ Decay

Introduction and Context

The positive mass theorem (PMT) is a foundational result in differential geometry and general relativity, asserting that an asymptotically flat $3$-manifold with non-negative scalar curvature must have non-negative ADM mass, with rigidity in the sense that zero ADM mass characterizes Euclidean space. Classical PMT arguments require the metric to converge to the Euclidean background at a sufficiently high ($C^2$) rate so the mass is well-defined. Recent advances, following Gromov and others, have sought to weaken the regularity and decay hypotheses, especially as scalar curvature exhibits features that are robust under $C^0$ perturbations despite being a second-order differential invariant.

This work addresses the rigidity statement in the positive mass theorem when only $C^0$ convergence at infinity is assumed, refining a conjecture by Gromov and confirming it in a near-optimal regime. The main result proves that rigidity holds under a quantitatively sharp $C^0$ decay rate, further illuminating the threshold at which the Euclidean metric is uniquely characterized by the decay and scalar curvature non-negativity alone.

Main Results

The central theorem establishes the following: If $g$ is a smooth Riemannian metric on $\mathbb{R}^3$ with non-negative scalar curvature and

$|g(x) - g_{\mathrm{euc}}(x)| = O(|x|^{-1-\tau})$

for some $\tau > 0$, then $3$0 is necessarily flat, i.e., isometric to Euclidean space. This rigorously confirms Gromov's rigidity conjecture (stated for $3$1 decay) for decay slightly stronger than $3$2.

Importantly, the authors show that decay exactly at the threshold $3$3 is sharp by constructing explicit counterexamples with non-flat metrics of non-negative scalar curvature having slower decay.

Analytical Approach

The proof blends harmonic function techniques with monotonicity formulae and functional analysis. The authors analyze Green's functions for the Laplacian with respect to $3$4 and exploit their asymptotic expansions:

• Existence and precise asymptotics of the Green's function $3$5 for the Laplacian in the metric $3$6 are established for the regime of interest, leveraging decay of the coefficients and elliptic regularity.
• A quantitative expansion is proven for $3$7 at infinity, valid in scale-invariant $3$8 norms, with leading behavior governed by the potential-theoretic structure of the Euclidean background and perturbative corrections controlled by the decay hypothesis.

A core feature of the argument is the use of Agostiniani–Mazzieri–Oronzio's $3$9-function, refined into a more stable, integrated $C^2$0-functional adapted to the $C^2$1 setting. The monotonicity and non-negativity of $C^2$2 under non-negative scalar curvature are established, paralleling classical monotonicity in PMT proofs.

The decay rate ensures that error terms contributing to $C^2$3 vanish at infinity. Thus, by monotonicity, $C^2$4 must be identically zero, which through established rigidity theorems for the $C^2$5-function implies flatness. The analysis relies on detailed control over elliptic solutions and scale-invariant functionals, exploiting precise decay estimates and the stability of the functionals to small perturbations in the metric.

Connections and Theoretical Implications

This work builds upon and sharpens a constellation of recent advances studying PMT and scalar curvature as $C^2$6 geometric notions [gromov2014dirac, bamler2016ricci, brendle2024scalar, mazurrowski2026quantification]. Prior extensions to low regularity, including synthetic definitions via Ricci flow and distributional notions of scalar curvature, have encountered challenges in generalizing the rigidity aspect of PMT, especially when defining mass in weak or $C^2$7-only settings [burkhardt2019pointwise, burkhardt2024adm]. The methodology here circumvents these difficulties by internalizing monotonicity into functionals well-defined in low regularity regimes, with the integrated $C^2$8-functional demonstrating robustness to $C^2$9-level decay.

This result elucidates the precise boundary in decay rates for PMT rigidity and provides a blueprint for future explorations involving even weaker or non-smooth convergence at infinity, possibly involving integral or distributional curvature bounds. The strategy also points to the effectiveness of harmonic and potential-theoretic techniques in scalar curvature questions where classical geometric measure theory may face difficulties due to low regularity.

Numerical and Structural Claims

• Rigidity established at $C^0$0 decay faster than $C^0$1:
  • For all $C^0$2, $C^0$3 decay suffices for flatness.
  • Explicit counterexample at the threshold $C^0$4 decay is constructed, showing sharpness of the result.
• The integrated $C^0$5-functional is proved non-negative and monotone; vanishing at infinity implies global flatness.

Future Directions

This work sharpens the conceptual boundary for positive mass rigidity and offers tools potentially applicable to compactness, limit, and stability properties of scalar curvature and mass under weak topologies. Possible future directions include:

• Extension to higher dimensions, where ADM mass and decay rates differ structurally, but similar questions about sharp decay for rigidity persist.
• Further analysis of isoperimetric, isocapacitary, or Ricci flow-based notions of mass in the $C^0$6 regime, particularly examining whether monotonicity-driven rigidity holds more generally.
• Development of analogous results for synthetic scalar curvature lower bounds and their interactions with geometric flows or weak convergence frameworks.

Conclusion

The paper rigorously confirms the rigidity of the positive mass theorem under quantitatively optimal $C^0$7 decay, verifying Gromov's conjecture in the near-optimal regime. The result combines advanced harmonic function techniques with monotonicity principles, demonstrating that flatness follows from scalar curvature non-negativity and $C^0$8 metric decay at infinity. This establishes a new regularity threshold for classical geometric rigidity, providing a foundation for further investigation of scalar curvature and mass quantization in low-regularity or synthetic geometric settings.

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References:

"Rigidity in the Positive Mass Theorem with $C^0$9 Decay" (2605.29915) Agostiniani, Mazzieri, Oronzio, "A Green’s function proof of the positive mass theorem" [agostiniani2024green] Gromov, "Four lectures on scalar curvature" [gromov2019four] Bamler, "A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature" [bamler2016ricci] Mazurowski, Yao, "Quantification of $C^0$0 Convergence in Dimension Three" (Mazurowski et al., 15 Apr 2026)