| | |
Summary: manuscripta math. 101, 143152 (2000) © Springer-Verlag 2000
Xianzhe Dai · Peter Petersen · Guofang Wei
Integral pinching theorems
Received: 17 February 1999
Abstract. Using Hamilton's Ricci flow we shall prove several pinching results for integral
curvature. In particular, we show that if p > n/2 and the Lp norm of the curvature tensor is
small and the diameter is bounded, then the manifold is an infra-nilmanifold. We also obtain
a result on deforming metrics to positive sectional curvature.
1. Introduction
The goal of this note is to prove several pinching results for manifolds with integral
curvature bounds. Integral pinching has been studied extensively in [3], [2], [12],
[13], [15], [8]. One distinct feature in our work is that assumptions on curvature
are entirely in terms of integral bounds and no assumption on volume, injectivity
radius or Sobolev constant is made.
Let us fix some notation before we state the results. For a Riemannian manifold
(M, g), we will denote by sec : M R the minimum of the sectional curvature
at each point, by R : 2T M 2T M the curvature operator, by Ric the Ricci
curvature, and by Scal the scalar curvature.
For functions and tensors we shall consistently use the normalized Lp norm
defined by
|
