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Let $M^n$ be a complete, open Riemannian manifold with $Ric geq 0$. In 1994, Grigori Perelman showed that there exists a constant $delta_{n}>0$, depending only on the dimension of the manifold, such that if the volume growth satisfies $alpha_M := lim_{r to infty} frac{Vol(B_p(r))}{omega_n r^n} geq 1-delta_{n}$, then $M^n$ is contractible. Here we employ the techniques of Perelman to find specific lower bounds for the volume growth, $alpha(k,n)$, depending only on $k$ and $n$, which guarantee the individual $k$-homotopy group of $M^n$ is trivial.
In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at
Suppose $(M,g)$ is a Riemannian manifold having dimension $n$, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity $C(X)$ is an Euclidean cone over the cross-section $X$.
We survey the results on fundamental groups of open manifolds with nonnegative Ricci curvature. We also present some open questions on this topic.
In a previous paper, we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimension $ge 6$. The purpose of the present paper is to use a different way
We are concerned with the study of different notions of curvature on graphs. We show that if a graph has stronger inner-outer curvature growth than a model graph, then it has faster volume growth too. We also study the relationhips of volume growth w