arXiv subjects

Michael Munn

Publications and source records attributed to Michael Munn.

24 records · Page 2Linked to original sources

arXiv2014

Alexandrov spaces with large volume growth

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 volume 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.

math.MG↗

arXiv2014

On the Size of a Ricci Flow Neckpinch via Optimal Transport

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 the 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.

math.MG↗

arXiv2012

Super Ricci flow for disjoint unions

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-Topping, 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.

math.DG↗

arXiv2010

Volume growth and the topology of Gromov-Hausdorff limits

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. Cheeger and Colding \cite{ChCoI} showed that given such a sequence of Riemannian manifolds it is possible to define a measure $ν$ 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 $ν$. 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 $\lim_{r \to \infty} \frac{ν(B_p(r))}{ω_n r^n} > α(k,n)$, then the $k$-th group of $(Y,p)$ is trivial. The constants $α(k,n)$ 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 $(Y,p)$.

math.MG↗

arXiv2009

Volume growth and the topology of manifolds with nonnegative Ricci curvature

Let $M^n$ be a complete, open Riemannian manifold with $\Ric \geq 0$. In 1994, Grigori Perelman showed that there exists a constant $δ_{n}>0$, depending only on the dimension of the manifold, such that if the volume growth satisfies $α_M := \lim_{r \to \infty} \frac{\Vol(B_p(r))}{ω_n r^n} \geq 1-δ_{n}$, then $M^n$ is contractible. Here we employ the techniques of Perelman to find specific lower bounds for the volume growth, $α(k,n)$, depending only on $k$ and $n$, which guarantee the individual $k$-homotopy group of $M^n$ is trivial.

math.DG↗

arXiv2008

On the appearance of Eisenstein series through degeneration

Let $Γ$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $\mathbb H$, and let $M = Γ\backslash \mathbb H$ be the associated finite volume hyperbolic Riemann surface. If $γ$ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If $γ$ 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 $γ\in Γ$ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.

math.NT↗