On Einstein metrics, normalized Ricci flow and smooth structures on $3\mathbb{CP}^2 # k \bar{\mathbb{CP}}^2$

TL;DR

This paper investigates the existence of Einstein metrics and non-singular normalized Ricci flow solutions on certain 4-manifolds, revealing how smooth structures influence geometric properties and extending previous topological and geometric analysis.

Contribution

It provides new results on Einstein metrics and Ricci flow behavior on specific 4-manifolds with different smooth structures, combining topological, geometric, and flow analysis methods.

Findings

Existence and non-existence results for Einstein metrics on specified 4-manifolds.

Obstructions to non-singular normalized Ricci flow solutions on exotic smooth structures.

Extension of previous topological and geometric techniques to new classes of 4-manifolds.

Abstract

In this paper, first we consider the existence and non-existence of Einstein metrics on the topological 4-manifolds $3\mathbb{CP}^2 # k \bar{\mathbb{CP}}^2$ (for $k \in {11, 13, 14, 15, 16, 17, 18}$) by using the idea of R\u{a}sdeaconu and \c{S}uvaina (2009) and the constructions in Park, Park, and Shin (arXiv:0906.5195v2) and in Park, Park, and Shin (2009). Then, we study the existence or non-existence of non-singular solutions of the normalized Ricci flow on the exotic smooth structures of these topological manifolds by employing the obstruction developed in Ishida (2008).