 
Summary: Comment. Math. Helv. 74 (1999) 345363
00102571/99/03034519 $ 1.50+0.20/0
c 1999 Birkh¨auser Verlag, Basel
Commentarii Mathematici Helvetici
Controlled Geometry via Smoothing
Peter Petersen, Guofang Wei and Rugang Ye§
Abstract. We prove that Riemannian metrics with a uniform weak norm can be smoothed
to having arbitrarily high regularity. This generalizes all previous smoothing results. As a
consequence we obtain a generalization of Gromov's almost flat manifold theorem. A uniform
Betti number estimate is also obtained.
Mathematics Subject Classification (1991). 53C20.
Keywords. Smoothing, regularity of metric, curvature bounds.
1. Introduction
An ultimate goal in geometry is to achieve a classification scheme, using natural
geometric quantities to characterize the topological type or diffeomorphism type of
Riemannian manifolds. While this grand scheme seems to be an impossible dream,
its basic philosophy has been a driving force in many important developments in
Riemannian geometry. The sphere theorems and various topological finiteness
theorems are typical examples. These results are concerned with control of global
topology of manifolds, and a crucial point therein is to control, uniformly, the local
