THE ENDS OF MANIFOLDS WITH BOUNDED GEOMETRY LINEAR GROWTH AND FINITE FILLING AREA

THE ENDS OF MANIFOLDS WITH BOUNDED GEOMETRY, LINEAR GROWTH AND FINITE FILLING AREA LOUIS FUNAR AND RENATA GRIMALDI Abstract. We prove that simply connected open manifolds of bounded ge- ometry, linear growth and sub-linear filling growth (e.g. finite filling area) are simply connected at infinity. MSC: 53 C 23, 57 N 15. Keywords: Bounded geometry, linear growth, filling area growth, simple con- nectivity at infinity. 1. Introduction A ubiquitous theme in Riemannian geometry is the relationship between the geometry (e.g. curvature, injectivity radius) and the topology. In studying non- compact manifolds constraints come from the asymptotic behaviour of geometric invariants (e.g. curvature decay, volume growth) as functions on the distance from a base point. The expected result is the manifold tameness out of geometric con- straints. This is illustrated by the classical theorem of Gromov which asserts that a complete hyperbolic manifold of finite volume and dimension at least 4 is the interior of a compact manifold with boundary. Our main result below yields tame- ness in the case when the filling area is finite, for those manifolds having bounded geometry and linear growth. We recall that: Definition 1.1. A non-compact Riemannian manifold has bounded geometry if the injectivity radius i is bounded from below and the absolute value of the curvature K is bounded from above.

• compact manifold


• linear growth


• riemannian manifold


• filling area


• bounded geometry


• result below