Manifolds with quadratic curvature decay and fast volume growth (Q1566352)

Let \(M\) be a complete connected non-compact Riemannian manifold. Fix the basepoint \(P_0\) and let \(S_r\) be the distance sphere of points of radius \(r\) about \(P_0\). One says \(M\) has \textit{quadratic curvature decay} if there exists \(C>0\) so that \(\limsup_{r\rightarrow\infty} \sup_{P\in S_r} r^2| K(P)| \leq C\) where \(K(P)\) denotes the maximal sectional curvature of a \(2\) plane in \(T_PM\). The author shows that if the constant \(C\) is small enough, if one has pinched Euclidean volume growth, and if \(M\) is noncollapsed at infinity, then \(M\) has finite topological type with ends that are cones over spherical space forms. The author notes that a result of B. Kleiner shows there is a surface of infinite topological type which admits noncollapsing metrics of roughly Euclidean volume growth and arbitrarily pinched quadratic curvature decay. The author also presents a result in which the pinched Euclidean volume growth assumption is replaced by a large scale convexity assumption.