Scalar curvature under weak limits of manifolds

Abstract: We show that scalar curvature lower bounds are preserved under certain weak convergence of smooth three manifolds to a smooth limit. More precisely, suppose that $M_k$ and $M$ are smooth, closed, Riemannian three manifolds. Assume that there are smooth, surjective, $λ_k$-Lipschitz maps $f_k\colon M_k \to M$ and that $\text{Vol}(M_k)\to \text{Vol}(M)$ and $λ_k\to 1$. Then if each $M_k$ has scalar curvature bounded below by $κ$ so does $M$. This result answers questions of Gromov, Sormani, Allen, and others. The proof relies on a delicate comparison between $μ$-bubbles in $M_k$ and $μ$-bubbles in $M$.

• The paper proves that, under surjective Lipschitz maps and volume convergence, scalar curvature lower bounds are preserved in weak limits of closed three-manifolds.
• The proof leverages variational methods using stable μ-bubbles and geometric measure theory to compare minimizers across manifolds.
• The result resolves conjectures by Gromov and Sormani, offering new insights into rigidity and scalar curvature stability in three-dimensional geometry.

Scalar Curvature Preservation under Weak Limits of Three-Manifolds

Introduction and Motivation

The preservation of scalar curvature lower bounds under various notions of geometric convergence plays a pivotal role in global differential geometry and geometric analysis, deeply influencing the study of rigidity and stability properties in scalar curvature geometry. While Alexandrov and CD-theory provide robust frameworks for the behavior of sectional and Ricci curvatures under limit processes, scalar curvature has eluded similarly general inheritance results except in special regimes. The paper "Scalar curvature under weak limits of manifolds" (2605.03136) provides a significant advance by establishing scalar curvature preservation under a new, notably weaker, convergence paradigm for closed three-dimensional Riemannian manifolds.

Main Result and Prior Context

The primary theorem resolves longstanding conjectures of Gromov and Sormani regarding the stability of scalar curvature under weak geometric convergence. Specifically, consider smooth, closed Riemannian 3-manifolds $\{M_k\}$ and a smooth, closed 3-manifold $M$. Assume:

• There exist smooth, surjective, $\lambda_k$-Lipschitz maps $f_k \colon M_k \to M$,
• $$\operatorname{Vol}(M_k) \rightarrow \operatorname{Vol}(M)$$,
• $\lambda_k \rightarrow 1$.

If $\operatorname{Scal}_{M_k} \geq \kappa$ for all $k$, then $\operatorname{Scal}_M \geq \kappa$.

This confirms Gromov's $(\alpha, \beta, \lambda)$-convergence conjecture in the three-dimensional, smooth setting, under the additional (and, as shown, essential) assumption of volume convergence [gromov2019four] [sormani2023conjectures] [allen2024oberwolfach].

The result is notably strong in light of previous counterexamples: scalar curvature lower bounds are not, in general, preserved under Gromov–Hausdorff, intrinsic flat, or distance function convergence [gromov2019four] [lee2024metric]. Thus, requiring surjective maps that become asymptotically isometric, together with volume convergence, is both necessary and—thanks to this result—sufficient.

Relationship to Notions of Convergence and Previous Work

The theorem establishes a form of curvature stability markedly weaker than $M$0-convergence of metrics (where scalar curvature preservation is classical [gromov2014dirac] [bamler2016ricci] [mazurowski2026quantification]), yet still strong enough to avoid known pitfalls of metric space convergence. It subsumes the "volume above distance below" convergence notion introduced by Allen, Perales, and Sormani [allen2024volume], and provides a positive resolution to several open questions posed in [sormani2023conjectures]. Importantly, the framework does not require the $M$1 to be diffeomorphic to $M$2, only the existence of surjective Lipschitz maps with controlled distortion and volume.

Technical Framework: Currents, Varifolds, $M$3-Bubbles

The proof leverages deep tools from geometric measure theory, including flat chains with mod 2 coefficients (currents), varifolds for capturing weak convergence of hypersurfaces, and Caccioppoli sets for handling nonsmooth boundaries. A core innovation is the careful comparison between stable $M$4-bubbles—generalized prescribed mean curvature hypersurfaces—in the approximating manifolds $M$5 and the limit manifold $M$6.

The authors construct variational problems on both $M$7 and $M$8 using functionals of the form

$M$9

minimized among Caccioppoli sets differing from a standard domain $\lambda_k$0 inside a controlled region. The existence and regularity theory for stable $\lambda_k$1-bubbles is imported from [gromov2019four] [zhu2021width] [zhou2019cmc] [zhou2020pmc].

Outline of the Proof

The proof is by contradiction: suppose scalar curvature in $\lambda_k$2 drops strictly below the common lower bound $\lambda_k$3 somewhere.

• A small geodesic sphere $\lambda_k$4 around such a point is shown to be the unique minimizer for an appropriately tuned $\lambda_k$5-bubble functional in $\lambda_k$6.
• This functional is lifted to $\lambda_k$7 via the surjective maps $\lambda_k$8.
• Minimizing $\lambda_k$9-bubbles $f_k \colon M_k \to M$0 are selected in each $f_k \colon M_k \to M$1, with strong variational properties and explicit estimates on area and volume.
• Pushforward analysis, area and co-area formulas, and careful multiplicity tracking establish weak convergence (in the varifold sense) of the pushforward of $f_k \colon M_k \to M$2 to $f_k \colon M_k \to M$3 and matching of various integral quantities.
• The stability inequalities for $f_k \colon M_k \to M$4 and for $f_k \colon M_k \to M$5 are compared. By delicately partitioning the integration region, exploiting sign conditions, and leveraging $f_k \colon M_k \to M$6-Lipschitz bounds, the authors derive a contradiction unless the scalar curvature lower bound passes to the limit.

Throughout, the necessity of volume convergence is highlighted: without this, the method fails due to possible mass loss in the limit. The degree-theoretic properties of the maps $f_k \colon M_k \to M$7 (asymptotically degree one modulo two) play a crucial role in pushforward arguments and in establishing proper limiting behavior.

Implications, Applications, and Future Directions

Practical and Theoretical Implications:

The result provides a rigorous and minimal set of hypotheses under which geometric PDE and scalar curvature rigidity phenomena are preserved in non-classical limits of Riemannian metrics. It justifies the use of volume/area-based variational arguments in a much broader context than previously recognized and opens the door for new rigidity and stability theorems in scalar curvature.

Connections to Open Problems:

This theorem resolves several major conjectures on scalar curvature stability, but leaves open the possibility of generalizations:

• Higher-dimensional extensions: The techniques crucially use three-dimensional arguments (e.g., properties of stable $f_k \colon M_k \to M$8-bubbles and the structure of scalar curvature in three-manifolds). Whether analogous results hold in higher dimensions, possibly with stronger regularity or structural hypotheses, remains an active topic.
• Spectral versions: The authors explicitly raise the question of spectral stability under $f_k \colon M_k \to M$9 convergence, i.e., whether first eigenvalues of Schrödinger-type operators involving scalar curvature are upper-semicontinuous in the same way.
• Refinements of convergence: The necessity and sufficiency of each hypothesis (e.g., volume convergence, surjectivity, Lipschitz control) merit further exploration. Even slight relaxations can lead to counterexamples [lee2024metric].

Applicability to Geometric Stability:

This generalizes the class of permissible degenerations in geometric analysis, augmenting the toolkit for studying moduli spaces of metrics with positive (or bounded) scalar curvature, and is anticipated to impact developments in scalar curvature comparison, positive mass theorems, and Penrose-type inequalities [mazurowski2025monotone] [dong2025stability] [cabrera2020stability].

Numerical Strength and Limitations

The main result is not accompanied by sharp quantitative estimates but rather establishes binary preservation of the scalar curvature lower bound. It provides a clear, logically complete answer to previously open conjectures under optimal regularity and convergence assumptions in dimension three. The negative results for weaker convergence highlight the sharpness of the hypotheses.

Conclusion

The paper achieves a definitive advance in the understanding of scalar curvature behavior under weak convergence of closed three-manifolds, giving a complete answer for the class of surjective, asymptotically isometric maps with volume convergence. By resolving conjectures of Gromov and Sormani, it clarifies the landscape of scalar curvature stability and sets the stage for further developments in the analytic and topological geometry of scalar curvature. Extensions to higher dimensions, weaker forms of convergence, and spectral versions remain as significant avenues for future research.

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References

• Mazurowski, L. & Yao, X. "Scalar curvature under weak limits of manifolds" (2605.03136).
• Gromov, M. "Four lectures on scalar curvature" (Gromov, 2019).
• Sormani, C. et al. "Conjectures on convergence and scalar curvature" [sormani2023conjectures].
• Allen, B. "Oberwolfach report: Scalar curvature stability" [allen2024oberwolfach].
• Allen, B., Perales, R., Sormani, C. "Volume above distance below" [allen2024volume].
• Lee, M.-C. & Topping, P. "Metric limits of manifolds with positive scalar curvature" [lee2024metric].
• Bamler, R. "A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature" [bamler2016ricci].
• Zhu, J. "Width estimate and doubly warped product" [zhu2021width].
• Zhou, X., Zhu, J.J. "Min-max theory for constant mean curvature hypersurfaces" [zhou2019cmc].
• Cabrera Pacheco, A.J., Ketterer, C., Perales, R. "Stability of graphical tori with almost nonnegative scalar curvature" [cabrera2020stability].