G-monopole classes, Ricci flow, and Yamabe invariants of 4-manifolds

2012/05/17 by Chanyoung Sung, Sung, Chanyoung
Mathematics · #53C44 #57M60 #57R57 #58E40 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG #msc:53C44 #msc:57M60 #msc:57R57 #msc:58E40

paper · pdf · doi:10.48550/arxiv.1205.3871

openalex publication_date 2012/05/17 · arxiv created 2013/03/13 · arxiv updated 2013/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

On a smooth closed oriented 4-manifold M with a smooth action by a finite group G, we show that a G-monopole class gives the L²-estimate of the Ricci curvature of a G-invariant Riemannian metric, and derive a topological obstruction to the existence of a G-invariant nonsingular solution to the normalized Ricci flow on M. In particular, for certain m and n, m\Bbb CP₂ # n\Bbb CP₂ admits an infinite family of topologically equivalent but smoothly distinct non-free actions of \Bbb Z_d such that it admits no nonsingular solution to the normalized Ricci flow for any initial metric invariant under such an action, where d>1 is a non-prime integer. We also compute the G-Yamabe invariants of some 4-manifolds with G-monopole classes and the oribifold Yamabe invariants of some 4-orbifolds.

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