L^p Ricci curvature pinching theorems for conformally flat Riemannian manifolds

Hong-Wei Xu and En-Tao Zhao

Vol. 245 (2010), No. 2, 381–396

DOI: 10.2140/pjm.2010.245.381

Abstract

Let M be an n-dimensional complete locally conformally flat Riemannian manifold with constant scalar curvature R and n ≥ 3. We first prove that if R = 0 and the L^n∕2 norm of the Ricci curvature tensor of M is pinched in [0,C₁(n)), then M is isometric to a complete flat Riemannian manifold, which improves Pigola, Rigoli, and Setti’s pinching theorem. Next, we prove that if n ≥ 6, R≠0, and the L^n∕2 norm of the trace-free Ricci curvature tensor of M is pinched in [0,C₂(n)), then M is isometric to a space form. Finally, we prove an Lⁿ trace-free Ricci curvature pinching theorem for complete locally conformally flat Riemannian manifolds with constant nonzero scalar curvature. Here C₁(n) and C₂(n) are explicit positive constants depending only on n.

Keywords

conformally flat manifold, rigidity, Ricci curvature tensor, L^p pinching problem, space form

Mathematical Subject Classification 2000

Primary: 53C20

Secondary: 53C25

Milestones

Received: 18 December 2008

Accepted: 19 May 2009

Published: 1 April 2010