Communications in Analysis and Geometry

Volume 25 (2017)

Number 1

The diffeomorphism type of manifolds with almost maximal volume

Pages: 243 – 267

DOI: http://dx.doi.org/10.4310/CAG.2017.v25.n1.a8

Authors

Curtis Pro (Department of Mathematics, University of Notre Dame, Indiana, U.S.A.)

Michael Sill (Department of Natural and Mathematical Sciences, California Baptist University, Riverside, Calif., U.S.A.)

Frederick Wilhelm (Department of Mathematics, University of California at Riverside)

Abstract

The smallest $r$ so that a metric $r$–ball covers a metric space $M$ is called the radius of $M$. The volume of a metric $r$-ball in the space form of constant curvature $k$ is an upper bound for the volume of any Riemannian manifold with sectional curvature $\geq k$ and radius $\leq r$. We show that when such a manifold has volume almost equal to this upper bound, it is diffeomorphic to a sphere or a real projective space.

Keywords

diffeomorphism stability, Alexandrov geometry

2010 Mathematics Subject Classification

53C20

Full Text (PDF format)

Received 14 November 2013

Published 9 June 2017