The Long-Time Behavior of the Ricci Tensor Under The Ricci Flow

2012/04/30 by Christian Hilaire, Hilaire, Christian
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1204.6733

openalex publication_date 2012/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that, given an immortal solution to the Ricci flow on a closed manifold with uniformly bounded curvature and diameter, the Ricci tensor goes to zero as t goes to infinity. We also show that if there exists an immortal solution on a closed 3-dimensional manifold such that the product of the square of the diameter with the norm of the curvature tensor is uniformly bounded, then the solution must be of type III.

Citations

• A Compactness Property for Solutions of the Ricci Flow

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• Curvature tensor under the Ricci flow
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