Spectral comparison and splitting theorems for the infinity-Bakry-Emery Ricci curvature

2025/09/27 by Wu, Jia-Yong
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.23182

Abstract

In this paper, we prove the diameter comparison, the global weighted volume comparison and the splitting theorem in weighted manifolds when the infinity-Bakry-Emery Ricci curvature has a lower bound in the spectrum sense. Our results extend Antonelli-Xu's spectral Bonnet-Myers and Bishop-Gromov theorems, and Antonelli-Pozzetta-Xu's spectral splitting theorem to weighted manifolds. Our results are also some supplements of Chu-Hao's spectral diameter and global volume comparisons, and Yeung's spectral splitting theorem in weighted manifolds.

Citations

• A Spectral Splitting Theorem for the N-Bakry Émery Ricci tensor
• A splitting theorem for manifolds with spectral nonnegative Ricci curvature and mean-convex boundary
• Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces
• Dimension constraints in some problems involving intermediate curvature
• Generalized soap bubbles and the topology of manifolds with positive scalar curvature
• Rigidity results for complete manifolds with nonnegative scalar curvature
• Width estimate and doubly warped product
• Four Lectures on Scalar Curvature
• Proof of Bishop's volume comparison theorem using singular soap bubbles
• 3-manifolds with nonnegative Ricci curvature
• Regularity of sets with quasiminimal boundary surfaces in metric spaces
• Comparison Geometry for the Bakry-Emery Ricci Tensor

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• Splitting theorems on complete manifolds with Bakry-Émery curvature
• Spectral comparison results for the N-Bakry-Emery Ricci tensor
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• New Volume Comparison results and Applications to degeneration of Riemannian metrics
• Laplacian comparison theorem on Riemannian manifolds with modified m-Bakry-Emery Ricci lower bounds for m≤1