Smooth Structures and Normalized Ricci Flows on Non-Simply Connected Four-Manifolds

2008/08/03 by Ishida, Masashi, Suvaina, Ioana
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.0808.0316

Abstract

A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors, we study the existence or non-existence of non-singular solutions of the normalized Ricci flow on 4-manifolds with non-trivial fundamental group and the relation with the smooth structures. For example, we prove that, for any finite cyclic group \mathbb Z_d, where d>1, there exists a compact topological 4-manifold X with fundamental group \mathbb Z_d, which admits at least one smooth structure for which non-singular solutions of the normalized Ricci flow exist, but also admits infinitely many distinct smooth structures for which \it no non-singular solution of the normalized Ricci flow exists. Related non-existence results on non-singular solutions are also proved. Among others, we show that there are no non-singular \ZZ_d-equivariant solutions to the normalized Ricci flow on appropriate connected sums of \bcp ²s and \cpb s (d>1).

Related

• The Normalized Ricci Flow on Four-Manifolds and Exotic Smooth Structures
• On Normalized Ricci Flow and Smooth Structures on Four-Manifolds with b⁺=1
• Non-singular solutions to the normalized Ricci flow equation
• Ricci flow on locally homogeneous closed 4-manifolds
• Non-singular solutions of normalized Ricci flow on noncompact manifolds of finite volume