Gauge theoretical approach to Einstein metrics and Ricci flow

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We proved new obstructions to the existence of Einstein metricsand non-singular solutions to normalized Ricci flow on 4-manifolds by using Seiberg-Wittenmonopole equations. As some applications of these new obstructions to 4-dimensionalgeometry, we proved a new existence theorem of 4-manifolds without Einstein metrics. Wealso proved the existence of 4-manifolds which admit no Einstein metric, but satisfy theHitchin-Thorpe inequality with volume entropy term. Moreover, we were able to prove thatthere exist infinitely many 4-manifolds which cannot admit non-singular solutions tonormalized Ricci flow, but satisfy Hitchin-Thorpe type inequality.