In this paper we address the relationship between Gromov-Hausdorff limits and intrinsic flat limits of complete Riemannian manifolds. In cite{SormaniWenger2010, SormaniWenger2011}, Sormani-Wenger show that for a sequence of Riemannian manifolds with
nonnegative Ricci curvature, a uniform upper bound on diameter, and non-collapsed volume, the intrinsic flat limit exists and agrees with the Gromov-Hausdorff limit. This can be viewed as a non-cancellation theorem showing that for such sequences, points dont cancel each other out in the limit. Here we prove a similar no-cancellation theorem, replacing the assumption of nonnegative Ricci curvature with a two-sided bound on Ricci curvature. This version corrects a mistake in the previous version of this paper (where we assume only an arbitrary lower Ricci bound) which was due to a crucial error in one of our supporting theorems for that argument.
Let $(X,d)$ be an $n$-dimensional Alexandrov space whose Hausdorff measure $mathcal{H}^n$ satisfies a condition giving the metric measure space $(X,d,mathcal{H}^n)$ a notion of having nonnegative Ricci curvature. We examine the influence of large vol
ume growth on these spaces and generalize some classical arguments from Riemannian geometry showing that when the volume growth is sufficiently large, then $(X,d,mathcal{H}^n)$ has finite topological type.
In this paper we apply techniques from optimal transport to study the neckpinch examples of Angenent-Knopf which arise through the Ricci flow on $mathbb{S}^{n+1}$. In particular, we recover their proof of single-point pinching along the flow. Using t
he methods of optimal transportation, we are able to remove the assumption of reflection symmetry for the metric. Our argument relies on the heuristic for weak Ricci flow proposed by McCann-Topping which characterizes super solutions of the Ricci flow by the contractivity of diffusions.
In this paper we consider compact, Riemannian manifolds $M_1, M_2$ each equipped with a one-parameter family of metrics $g_1(t), g_2(t)$ satisfying the Ricci flow equation. Motivated by a characterization of the super Ricci flow developed by McCann-T
opping, we introduce the notion of a super Ricci flow for a family of distance metrics defined on the disjoint union $MM$. In particular, we show such a super Ricci flow property holds provided the distance function between points in $M_1$ and $M_2$ evolves by the heat equation. We also discuss possible applications and examples.
We examine topological properties of pointed metric measure spaces $(Y, p)$ that can be realized as the pointed Gromov-Hausdorff limit of a sequence of complete, Riemannian manifolds ${(M^n_i, p_i)}_{i=1}^{infty}$ with nonnegative Ricci curvature. Ch
eeger and Colding cite{ChCoI} showed that given such a sequence of Riemannian manifolds it is possible to define a measure $ u$ on the limit space $(Y, p)$. In the current work, we generalize previous results of the author to examine the relationship between the topology of $(Y, p)$ and the volume growth of $ u$. In particular, we prove a Abresch-Gromoll type excess estimate for triangles formed by limiting geodesics in the limit space. Assuming explicit volume growth lower bounds in the limit, we show that if $lim_{r to infty} frac{ u(B_p(r))}{omega_n r^n} > alpha(k,n)$, then the $k$-th group of $(Y,p)$ is trivial. The constants $alpha(k,n)$ are explicit and depend only on $n$, the dimension of the manifolds ${(M^n_i, p_i)}$, and $k$, the dimension of the homotopy in $(Y,p)$.
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.
Let $Gamma$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $mathbb H$, and let $M = Gamma backslash mathbb H$ be the associated finite volume hyperbolic Riemann surface. If $gamma$ is parabolic, there is an associated
(parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If $gamma$ is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If $gamma in Gamma$ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.
