Collapsing in the L² curvature flow

2012/01/05 by Jeff Streets, Streets, Jeff
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1201.1266

openalex publication_date 2012/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show some results for the L² curvature flow linked by the theme of addressing collapsing phenomena. First we show long time existence and convergence of the flow for SO(3)-invariant initial data on S³, as well as a long time existence and convergence statement for three-manifolds with initial L² norm of curvature chosen small with respect only to the diameter and volume, which are both necessary dependencies for a result of this kind. In the critical dimension n = 4 we show a related low-energy convergence statement with an additional hypothesis. Finally we exhibit some finite time singularities in dimension n ≥ 5, and show examples of finite time singularities in dimension n ≥ 6 which are collapsed on the scale of curvature.

Citations

• Fourth order curvature flows and geometric applications
• The gradient flow of the L² curvature energy near the round sphere
• Ricci flow with surgery on three-manifolds
• The long time behavior of fourth-order curvature flows

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