The ends of manifolds with bounded geometry, linear growth and finite filling area (Q1826907)

The paper concerns noncompact Riemannian manifolds having bounded geometry, i.e. for which the injectivity radius \(i\) is bounded from below and the absolute value of the curvature \(K\) is bounded from above. Important class of manifolds of bounded geometry is formed by those with linear volume growth [\textit{J. Cheeger} and \textit{M. Gromov}, ''Chopping Riemannian manifolds''. in: Differential geometry. A symposium in honour of Manfredo do Carmo, Proc. Int. Conf., Rio de Janeiro/Bras. 1988, Pitman Monogr. Surv. Pure Appl. Math. 52, 85--94 (1991; Zbl 0722.53045)]. The authors are mainly interested in those for which the filling growth is sublinear, i.e. \(\lim_{r\rightarrow \infty }\frac{F_{X}\left( l,r\right) }{r} =0\) where \(F_{X}\left( l,r\right) \) is the smallest area of the disk in \(X\) filling a loop of length \(l\) lying in the metric ball \( B_{X}\left( r\right) \) of radius \(r\) on \(X.\) The main result of the paper is the following theorem: Theorem. A simply connected open Riemannian manifold of bounded geometry, linear growth and sublinear filling growth is simply connected at infinity. We explain that by the space (a noncompact polyhedron) of simply connected at infinite the authors mean a space \(X\) for which for a given compact set \( K\subset X\) there exists another compact set \(L\) with \(K\subset L\subset X,\) such that the map induced by the inclusion \(\pi _{1}\left( X\backslash L\right) \rightarrow \pi _{1}\left( X\backslash K\right) \) is trivial. In consequence, the authors have obtained the following topological characterization: An open contractible Riemannian manifold of bounded geometry, linear growth and sub-linear filling growth is diffeomorphic (only homeomorphic in dimension 4) to the Euclidean space (one needs to assume the irreducibility in dimension 3). An example of Riemannian metric \(g\) of bounded geometry on \(\mathbb{R}^{n}\) (for \(n\geq 3\) ) such that \(\lim_{r\rightarrow \infty }\frac{F_{\left( \mathbb{R}^{n},g\right) }\left( l,r\right) }{r}\neq 0\) is given.