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

Abstract: A quantitative version of the scalar lower bound under $C^0$ convergence was conjectured by Gromov. More recently, Mazurowski and Yao proved that a refined form of Gromov's conjecture holds in dimension three. Furthermore, they constructed examples demonstrating that such a refinement is necessary. In this paper, we establish that the refined quantitative bound holds in all dimensions greater than or equal to three.

• The paper establishes sharp quantitative bounds for scalar curvature under C⁰ and Lᵖ convergence via Ricci flow smoothing.
• It demonstrates optimal exponent scaling through rigorous local analysis and explicit geometric constructions.
• The results extend curvature continuity to collapsing geometries, impacting the study of metric perturbations and moduli spaces.

Quantitative Scalar Curvature Bounds under $C^0$ and $L^p$ Convergence

Context and Motivation

The quantification of scalar curvature preservation under weak convergence of Riemannian metrics has remained a central topic since Gromov’s investigation of compactness and stability properties in scalar curvature geometry. The traditional qualitative result establishes that scalar curvature lower bounds are preserved under $C^0$ convergence, notably reformulated by Gromov in terms of $C^0$-stable quantities and further proved via Ricci flow smoothing methods [Gromov2014, Bamler2016, Burkhardt2019]. However, Gromov’s conjecture about quantitative preservation—specifically, how the scalar curvature lower bound degrades as two metrics converge in the $L^\infty$ norm—has been shown to require refinement, with Mazurowski and Yao demonstrating both the necessity and sharpness of the correct scaling in dimension $n \geq 3$ [MazurowskiYao2026].

This paper, "Quantification of scalar curvature under $C^0$ convergence using smoothing" (2604.17759), systematically establishes refined quantitative bounds for scalar curvature in all dimensions $n \geq 3$, employing Ricci flow smoothing and local analysis. The results generalize prior partial dimension-specific results and connect scalar curvature quantification to local $L^p$ closeness, providing sharp exponent bounds and broad applicability, including collapsing geometries.

Main Results

Quantitative Scalar Curvature Bounds

The principal theorem asserts that for $n \geq 3$, if two smooth metrics $L^p$0 and $L^p$1 are $L^p$2-close on an intrinsic domain $L^p$3, then

$L^p$4

where $L^p$5 is a dimensional constant and $L^p$6 encodes the domain scale. The result extends to an $L^p$7 framework under a non-collapsing volume assumption: $L^p$8 with sharpness of the exponent established via explicit examples inspired by Mazurowski-Yao.

These bounds are fundamentally scaling-invariant and applicable even in the collapsing case, distinguishing them from classical results that require volume non-collapsing assumptions. The approach utilizes Ricci-DeTurck flows localized via diffeomorphisms and persistence of scalar curvature lower bound under Ricci flow smoothing, leveraging local maximum principles [LeeTam2022].

Preservation under Measure Convergence

The corollary derived from the quantitative bounds shows that the scalar curvature lower bound is preserved under $L^p$9 convergence in the sense of measure, provided the $C^0$0 closeness and uniform lower bounds for the sequence hold. This generalizes earlier qualitative results and provides quantitative control on the limit metric’s scalar curvature.

Sharpness via Explicit Construction

Appendix details show that the exponents in the main theorems are optimal, referencing Mazurowski-Yao’s construction of rotationally symmetric, conformal metrics, where the scalar curvature lower bound cannot be improved beyond the stated quantitative scaling.

Methodology

The methodology is built on the stability theory of the Ricci-DeTurck flow, enabling comparison between local smoothing of the background metric $C^0$1 and its perturbation $C^0$2. The Ricci-DeTurck framework is strictly parabolic, allowing for fine control of local geometry under smoothing, and ensuring that the essential geometric information is preserved. The key technical ingredients include:

• Lifted normal coordinates and diffeomorphism construction: By comparing metrics via local charts (exp-maps), the paper transfers geometric control from the original manifold to (almost) Euclidean balls, where Ricci flows can be analyzed more effectively.
• Local scalar maximum principles: These enable precise preservation of scalar curvature lower bounds; the results are then transported back to the original manifold via the diffeomorphism.
• Local $C^0$3 improvement: Smoothing provides a mechanism for interpolation between $C^0$4 and $C^0$5 bounds, giving the exponent in the quantitative estimate for scalar curvature.
• Localization strategies: Curvature control is achieved via local analysis on domains with prescribed geometry, neither requiring global non-collapsing nor full Ricci flow compactness arguments.

Numerical and Theoretical Highlights

• Sharp exponent: The $C^0$6 exponent in $C^0$7 convergence and its higher-dimensional $C^0$8 counterpart are proven to be optimal, reflecting the degeneracy inherent in conformal deformation and local smoothing.
• Non-collapsing independence: The results hold in collapsing geometries, provided the domain geometry can be locally controlled (bounded curvature, intrinsic balls).
• Extensions: The methods allow generalization to scalar curvature continuity under quantitative geometric hypotheses, providing new tools to approach scalar curvature limits in singular geometries.

Implications and Prospects

Theoretical implications include a rigorous framework for quantitative scalar curvature compactness in varying dimensions, laying groundwork for further study of geometric convergence under weaker norms. The results provide sharp controls that are essential in applications involving metric convergence, particularly in settings where $C^0$9 or $C^0$0 perturbations are unavoidable, such as lower-regularity metric measure spaces or singular Ricci flow solutions.

Practically, these quantitative bounds facilitate precise estimates relevant for moduli spaces of scalar curvature-bounded metrics, metric collapse phenomena, and convergence of geometric flows with rough initial data.

Future directions may include:

• Extension to convergence scenarios weaker than $C^0$1, such as Gromov-Hausdorff convergence or intrinsic flat convergence, potentially requiring new smoothing techniques.
• Exploration of scalar curvature quantification in singular settings, e.g., metric measure spaces or Alexandrov spaces.
• Refinement of localization procedures for curvature bounds, possibly connecting with synthetic curvature lower bounds in non-smooth spaces (see [KazarasXu]).

Conclusion

This paper establishes precise, dimension-independent quantitative bounds for scalar curvature under $C^0$2 and $C^0$3 convergence, proving optimality of the exponents and extending previous partial dimension-specific results through Ricci flow smoothing and localized analysis. The results contribute important mathematical infrastructure for quantitative geometric analysis in scalar curvature, with broad implications for metric convergence and flow theory in Riemannian geometry.