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Complete 3-manifolds of positive scalar curvature with quadratic decay

Florent Balacheff, Teo Gil Moreno de Mora Sardà, Stéphane Sabourau

TL;DR

This work extends the 3-manifold PSC classification to complete, noncompact manifolds with PSC and at most quadratic decay by proving a topological decomposition into a (possibly infinite) connected sum of spherical manifolds and $S^2\times S^1$, under a decay threshold $C>64\pi^2$. A central innovation is replacing analytic μ-bubble techniques with a fill-radius–based, topological approach, deriving a curvature-free criterion that yields the same decomposition and showing the decay rate is optimal via the indecomposable example $\mathbb{R}^2\times\mathbb{S}^1$. The paper also proves a surgery-type result: any such decomposition admits a metric of uniformly positive scalar curvature, linking infinite connected sums to PSC geometry, while ruling out aspherical summands through ends arguments. Together, these results tie PSC geometry to fill radius growth, ends structure, and infinite connected-sum decompositions, providing sharp decay-rate constraints and robust topological conclusions.

Abstract

We prove that if an orientable 3-manifold $M$ admits a complete Riemannian metric whose scalar curvature is positive and has a subquadratic decay at infinity, then it decomposes as a (possibly infinite) connected sum of spherical manifolds and $\mathbb{S}^2 \times \mathbb{S}^1$ summands. This generalises a theorem of Gromov and Wang by using a different, more topological, approach. As a result, the manifold $M$ carries a complete Riemannian metric of uniformly positive scalar curvature, which partially answers a conjecture of Gromov. More generally, the topological decomposition holds without any scalar curvature assumption under a weaker condition on the filling discs of closed curves in the universal cover based on the notion of fill radius. Moreover, the decay rate of the scalar curvature is optimal in this decomposition theorem. Indeed, the manifold $\mathbb{R}^2 \times \mathbb{S}^1$ supports a complete metric of positive scalar curvature with exactly quadratic decay, but does not admit a decomposition as a connected sum.

Complete 3-manifolds of positive scalar curvature with quadratic decay

TL;DR

This work extends the 3-manifold PSC classification to complete, noncompact manifolds with PSC and at most quadratic decay by proving a topological decomposition into a (possibly infinite) connected sum of spherical manifolds and , under a decay threshold . A central innovation is replacing analytic μ-bubble techniques with a fill-radius–based, topological approach, deriving a curvature-free criterion that yields the same decomposition and showing the decay rate is optimal via the indecomposable example . The paper also proves a surgery-type result: any such decomposition admits a metric of uniformly positive scalar curvature, linking infinite connected sums to PSC geometry, while ruling out aspherical summands through ends arguments. Together, these results tie PSC geometry to fill radius growth, ends structure, and infinite connected-sum decompositions, providing sharp decay-rate constraints and robust topological conclusions.

Abstract

We prove that if an orientable 3-manifold admits a complete Riemannian metric whose scalar curvature is positive and has a subquadratic decay at infinity, then it decomposes as a (possibly infinite) connected sum of spherical manifolds and summands. This generalises a theorem of Gromov and Wang by using a different, more topological, approach. As a result, the manifold carries a complete Riemannian metric of uniformly positive scalar curvature, which partially answers a conjecture of Gromov. More generally, the topological decomposition holds without any scalar curvature assumption under a weaker condition on the filling discs of closed curves in the universal cover based on the notion of fill radius. Moreover, the decay rate of the scalar curvature is optimal in this decomposition theorem. Indeed, the manifold supports a complete metric of positive scalar curvature with exactly quadratic decay, but does not admit a decomposition as a connected sum.
Paper Structure (10 sections, 25 theorems, 48 equations, 8 figures)

This paper contains 10 sections, 25 theorems, 48 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Scalar curvature and fill radius
  3. Fill radius and simply connectedness at infinity on contractible curves
  4. Localised compression and exhaustion by compact domains with incompressible boundaries
  5. Fill radius and number of ends
  6. Fill radius and aspherical summands
  7. Proof of the decomposition theorem
  8. An indecomposable 3-manifold with quadratic decay
  9. The surgery theorem for infinite connected sums
  10. Statements and Declarations

Key Result

Theorem 1.1

Let $M$ be a complete orientable Riemannian 3-manifold with uniformly positive scalar curvature. Then $M$ decomposes as a possibly infinite connected sum of spherical manifolds and $\mathbb{S}^2 \times \mathbb{S}^1$.

Figures (8)

  • Figure 1: The neighbourhood $\Omega(s)$ in the stable minimal disc $\Sigma$.
  • Figure 2: Compressing a surface.
  • Figure 3: Deforming a loop.
  • Figure 4: Scheme of the proof of Lemma \ref{['le:Gromov_Lawson_Lemma']}.
  • Figure 5: Minimising curve $\tilde{\gamma}$ joining two points $y_1$ and $y_2$ lying in $\bar{X}$.
  • ...and 3 more figures

Theorems & Definitions (63)

  • Theorem 1.1: Gromov_2023Wang_2023
  • Definition 1.2
  • Theorem 1.3
  • Conjecture 1.4: Critical Rate of Decay Conjecture Gromov_2023
  • Corollary 1.5
  • Definition 1.6
  • Theorem 1.7: Gromov_Lawson_1983, Schoen_Yau_1983
  • Definition 1.8
  • Proposition 1.9
  • Theorem 1.10
  • ...and 53 more