[Paper Review] Geometry of positive scalar curvature on complete manifold

Bo Zhu|arXiv (Cornell University)|Jan 29, 2022

Geometric Analysis and Curvature Flows4 citations

TL;DR

This paper investigates the interplay between positive scalar curvature and geometric size in complete, non-compact manifolds with non-negative Ricci curvature. Using Gromov-Hausdorff convergence, Lipschitz mapping techniques, and curvature estimates, it establishes volume growth bounds and uniform diameter control on level sets of Busemann functions in 3D manifolds, proving that scalar curvature positivity restricts large-scale geometry even without compactness.

ABSTRACT

In this paper, we study the interplay of geometry and positive scalar curvature on a complete, non-compact manifold with non-negative Ricci curvature. In three-dimensional manifold, we prove a minimal volume growth, an estimate of integral of scalar curvature and width. In higher dimensional manifold, we obtain a volume growth with a stronger condition.

Motivation & Objective

• To understand how positive scalar curvature constrains large-scale geometry in complete, non-compact Riemannian manifolds with non-negative Ricci curvature.
• To address Yau’s open problem on the asymptotic behavior of the integral of scalar curvature relative to volume growth.
• To establish uniform diameter bounds on level sets of Busemann functions in 3-dimensional manifolds under positive scalar curvature and non-negative Ricci curvature.
• To investigate whether injectivity radius or conjugate radius bounds persist in the non-compact setting, extending results from closed manifolds.
• To provide a local estimate on the integral of scalar curvature in terms of conjugate radius and volume, under non-negative Ricci curvature.

Proposed method

• Utilizes Gromov-Hausdorff convergence of pointed sequences of manifolds to analyze asymptotic geometry.
• Applies a Lipschitz map construction from ℝⁿ to 𝕊ⁿ with small Lipschitz constant and non-zero degree to derive contradictions under scalar curvature bounds.
• Employs the Spherical Lipschitz Bound Theorem to rule out infinite injectivity radius or unbounded volume growth.
• Uses Busemann functions associated with rays to define level sets and analyzes their diameter behavior via pointed Gromov-Hausdorff limits.
• Applies monotonicity formulas and curvature comparison techniques from Colding and Minicozzi in the 3D case.
• Establishes a local integral estimate: ∫_{B(p,R−c)} Sc ≤ n(n−1)(π/c)² vol(B(p,R)) for c ≤ conj(M), under non-negative Ricci curvature.

Experimental results

Research questions

• RQ1 Does positive scalar curvature on a complete, non-compact manifold with non-negative Ricci curvature imply a uniform bound on the diameter of level sets of Busemann functions?
• RQ2 Can the volume growth of a complete, non-compact manifold with positive scalar curvature and non-negative Ricci curvature be bounded above by a power of radius?
• RQ3 Does the injectivity radius remain uniformly bounded above under positive scalar curvature and non-negative Ricci curvature in the non-compact setting?
• RQ4 Is there a local upper bound on the integral of scalar curvature in terms of volume and conjugate radius under non-negative Ricci curvature?
• RQ5 Does the asymptotic behavior of the integral of scalar curvature relative to volume growth satisfy Yau’s conjectured limsup condition?

Key findings

• In 3-dimensional complete manifolds with non-negative Ricci curvature and scalar curvature Sc ≥ 2, the diameter of any level set of a Busemann function is uniformly bounded above.
• The volume growth satisfies limsup_{R→∞} vol(B(p,R))/R^{n−1} < ∞ under positive scalar curvature and non-negative Ricci curvature, implying sublinear volume growth in higher dimensions.
• For 3-dimensional manifolds with two ends, the geometry is isometric to 𝕊²×ℝ, and the level sets of the projection function have uniformly bounded diameter.
• The injectivity radius is bounded above by π under positive scalar curvature and non-negative Ricci curvature, extending the maximal injectivity radius theorem to the non-compact case.
• A local estimate is proven: ∫_{B(p,R−c)} Sc ≤ n(n−1)(π/c)² vol(B(p,R)) for all c ≤ conj(M), under Ric(g) ≥ 0.
• The paper rules out the existence of complete, non-compact manifolds with Sc ≥ n(n−1), Ric ≥ 0, and infinite injectivity radius via contradiction using Lipschitz maps to spheres.