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

Research Overview

• 💡 Shuai Zhang from Tsinghua University presented research on the asymptotic volume growth ratio for 3-dimensional manifolds with positive scalar curvature.
• 🤝 This work is a joint effort with Guodong Wei and Guoyi Xu, along with some related results.

Historical Context and Prior Findings

• 📜 The discussion stems from Gromov's 1980s conjecture concerning 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.
• 🎯 Bo Zhu proved the finiteness of this ratio in 2022, albeit with additional hypotheses.
• 📌 For dimension 3, proofs were also provided by Ovidiu Munteanu and Jiaping Wang in 2022, and later by Otis Chodosh, Chao Li, and Douglas Stryker in 2023.

Speaker's Specific Contributions

• 🚀 The speaker's research achieved a sharp upper bound for this ratio in dimension 3, along with corresponding rigidity.
• 🛠️ This result was obtained using Cheeger-Colding's almost splitting theorem and the µ-bubble method.

Fundamental Geometric Principles

• 🔬 The Taylor expansion of the volume for geodesic balls shows that positive scalar curvature leads to a smaller volume in the local range.
• 📈 While the local formula is for small radii, non-negative Ricci curvature provides control over the volume for larger radii.
• ✅ Bishop-Gromov's comparison theorem indicates that the volume ratio of a geodesic ball to Euclidean space has a limit no greater than one, with the limit equaling one if and only if the manifold is isometric.