Convergence for Yamabe metrics of positive scalar curvature with integral bounds on curvature

Kazuo Akutagawa

Vol. 175 (1996), No. 2, 307–335

DOI: 10.2140/pjm.1996.175.307

Abstract

Let 𝒴₁(n,μ₀) be the class of compact connected smooth n-manifolds M (n ≥ 3) with Yamabe metrics g of unit volume which satisfy

where [g] and μ(M,[g]) denote the conformal class of g and the Yamabe invariant of (M,[g]), respectively. The purpose of this paper is to prove several convergence theorems for compact Riemannian manifolds in 𝒴₁(n,μ₀) with integral bounds on curvature. In particular, we present a pinching theorem for flat conformal structures of positive Yamabe invariant on compact 3-manifolds.

Mathematical Subject Classification 2000

Primary: 58E11

Secondary: 53C21

Milestones

Received: 24 October 1994

Published: 1 October 1996