Asymptotic Volume Growth Ratio on 3-Manifolds with Positive Scalar Curvature

Gromov's Conjecture and Background

• 💡 Gromov's conjecture (1980s) posits the finiteness of the (n-2)-order asymptotic volume growth ratio for geodesic balls on n-manifolds with positive scalar and non-negative Ricci curvature.
• 🧠 The Bishop-Gromov theorem shows that for non-negative Ricci curvature, the volume ratio is no greater than one, but its limit as radius approaches zero is always one, providing no information.
• 🚀 Taking the limit as radius tends to infinity (asymptotic limit) provides more meaningful information about the manifold's global structure.

Advances in Dimension 3

• ✅ For 3-dimensional manifolds, several proofs emerged in 2022-2023, including work by Munteanu & Wang, and Chodosh, Li & Stryker.
• 🛠️ These proofs often utilize techniques like the µ-bubble method and Cheeger-Colding's almost splitting theorem.
• 🎯 The speaker's joint work (2024) established a sharp upper bound of 8π for this ratio in dimension 3, with rigidity for the cylinder S² × R.

Manifold Structure and Rigidity

• 🧩 The theorem distinguishes between manifolds with one end (upper bound 4π) and those with at least two ends (upper bound 8π).
• 🔑 For manifolds with two or more ends, the Gromov-Colding splitting theorem implies they must split into a cylinder, leading to the 8π bound and rigidity.
• ⚠️ The cylinder S² × R is identified as the manifold with the maximal volume growth ratio in dimension 3.

Related Conjectures and Tools

• 🔍 A generalization by Yau considers integrating scalar curvature in geodesic balls, which remains largely unknown even for dimension three.
• 🔬 The Bonnet-Myers theorem and a lemma by Schoen-Yau (for 2D submanifolds) are crucial tools for estimating diameter bounds, particularly in the context of µ-bubble surfaces.
• 💡 Gromov's conjecture on microscopic dimension suggests that the microscopic dimension of n-manifolds with positive scalar curvature is no greater than n-2.

Proof Sketch for Volume Growth

• ✂️ The proof involves cutting the manifold into cylinder regions (or its universal cover) along geodesic spheres.
• 📏 The height of these cylinder regions is estimated using Cheeger-Colding's almost splitting theorem, which helps control the diameter of "arms."
• 📊 The area of each slice (geodesic sphere) is controlled using µ-bubble surfaces, leveraging the diameter upper bound for 2D surfaces, which is why the method is effective in dimension 3.