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# Communications in Analysis and Geometry

## Volume 25 (2017)

### Number 1

### The diffeomorphism type of manifolds with almost maximal volume

Pages: 243 – 267

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

#### Authors

#### 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

Received 14 November 2013

Published 9 June 2017