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Chris Connell

Publications and source records attributed to Chris Connell.

At least 19 recordsLinked to original sources
arXiv2025

On the Jacobian of the Douady-Earle extension

Given an isotopy class between two closed hyperbolic surfaces, the Douady--Earle extension provides a unique analytic diffeomorphism representative. In this paper we investigate the Jacobian of the Douady--Earle extension map $F$. We prove that $|\operatorname{Jac} F| \equiv 1$ precisely when $F$ is an isometry. Moreover, we construct a sequence of hyperbolic surfaces $\{\Sigma_i\}$ together with a fixed domain surface $\Sigma_0$ for which the Douady--Earle extension maps $F_i:\Sigma_0\to\Sigma_i$ satisfy $\max_{x\in\Sigma_0} \operatorname{Jac} F_i \to +\infty$.

math.GT↗
arXiv2025

Homological dimension of discrete subgroups in higher rank simple Lie groups

We investigate the homological dimensions of discrete subgroups of non-compact simple Lie groups using a flow recently introduced by the authors. Using new estimates on the critical index for a broad class of discrete subgroups, we prove that the homological dimension of non-lattice, discrete, Zariski-dense subgroups is bounded above by $n-\alpha r$ where $n$ is the dimension of the associated symmetric space, $r$ is the real rank, and $\alpha \geq \frac{1}{8}$. Under the assumption that the discrete group has regular limit cone, our bound improves to $\frac{5}{6}n+1$. Additionally, we make some conjectures on the possible homological dimensions for non-lattice discrete subgroups and pose a few questions in the more general semisimple setting.

math.GT↗
arXiv2024

Volume entropy and rigidity for RCD-spaces

We develop the barycenter technique of Besson--Courtois--Gallot so that it can be applied on RCD metric measure spaces. Given a continuous map $f$ from a non-collapsed RCD$(-(N-1),N)$ space $X$ without boundary to a locally symmetric $N$-manifold we show a version of BCG's entropy-volume inequality. The lower bound involves homological and homotopical indices which we introduce. We prove that when equality holds and these indices coincide $X$ is a locally symmetric manifold, and $f$ is homotopic to a Riemannian covering whose degree equals the indices. Moreover, we show a measured Gromov--Hausdorff stability of $X$ and $Y$ involving the homotopical invariant. As a byproduct, we extend a Lipschitz volume rigidity result of Li--Wang to RCD$(K,N)$ spaces without boundary. Finally, we include an application of these methods to the study of Einstein metrics on $4$-orbifolds.

math.DG↗
arXiv2024

Simplicial volume and isolated, closed totally geodesic submanifolds of codimension one

We show that for any closed Riemannian manifold with dimension at least two and with nonpositive curvature, if it admits an isolated, closed totally geodesic submanifold of codimension one, then its simplicial volume is positive. As a direct corollary of this, for any nonpositively curved analytic manifold with dimension at least three, if its universal cover admits a codimension one flat, then either it has non-trivial Euclidean de Rham factors, or it has positive simplicial volume.

math.GT↗
arXiv2024

Rigidity of compact rank one symmetric spaces

We consider rigidity properties of compact symmetric spaces $X$ with metric $g_0$ of rank one. Suppose $g$ is another Riemannian metric on $X$ with sectional curvature $\kappa$ bounded by $0 \leq \kappa \leq 1$. If $g$ equals $g_0$ outside a convex proper subset of $X$, then $g$ is isometric with $g_0$. We also exhibit examples of surfaces showing that the nonnegativity of the curvature is needed. Our main result complements earlier results on other symmetric spaces by Gromov and Schroeder-Ziller.

math.DG↗
arXiv2023

Nonpositively curved $4$-manifolds with zero Euler characteristic

We show that for any closed nonpositively curved Riemannian 4-manifold $M$ with vanishing Euler characteristic, the Ricci curvature must degenerate somewhere. Moreover, for each point $p\in M$, either the Ricci tensor degenerates or else there is a foliation by totally geodesic flat 3-manifolds in a neighborhood of $p$. As a corollary, we show that if in addition the metric is analytic, then the universal cover of $M$ has a nontrivial Euclidean de Rham factor. Finally we discuss how this result creates an implication of conjectures on simplicial volume in dimension four.

math.DG↗
arXiv2023

Boundary actions of lattices and $C^0$ local semi-rigidity

We consider actions of cocompact lattices in semisimple Lie groups of the noncompact type on their boundaries $G/Q$, $Q$ a parabolic group, the so-called standard actions. We show that perturbations of the standard action in the homeomorphism group continuously factor onto the original standard action by a semi-conjugacy close to the identity. This generalizes works by Bowden, Mann, Manning and Weisman in the setting of negative curvature or Gromov hyperbolic groups. Finally, we also construct perturbations of the action of lattices on the geodesic boundary which are not $C^0$ semi-conjugate to the original action.

math.DS↗
arXiv2023

The natural flow and the critical exponent

Inspired by work of Besson-Courtois-Gallot, we construct a flow called the natural flow on a non-positively curved Riemannian manifold $M$. As with the natural map, the $k$-Jacobian of the natural flow is directly related to the critical exponent $\delta$ of the fundamental group. There are several applications of the natural flow that connect dynamical, geometrical, and topological invariants of the manifold. First, we give $k$-dimensional linear isoperimetric inequalities when $k > \delta$. This, in turn, produces lower bounds on the Cheeger constant. We resolve a recent conjecture of Dey-Kapovich on the non-existence of $k$-dimensional compact, complex subvarieties of complex hyperbolic manifolds with $2k > \delta$. We also provide upper bounds on the homological dimension, generalizing work of Kapovich and work of Farb with the first two authors. Using the natural flow together with Morse theory, we also give upper bounds on the cohomological dimension, which partially resolve a conjecture of Kapovich. Finally, we introduce a new growth condition on the Bowen-Margulis measure that we call uniformly exponentially bounded that we connect to the cohomological dimension and which could be of independent interest.

math.DG↗
arXiv2021

Carnot metrics, Dynamics and Local Rigidity

This paper develops new techniques for studying smooth dynamical systems in the presence of a \CC metric. Principally, we employ the theory of Margulis-Mostow, M\'etivier, Mitchell and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth \CC metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter application we also prove structural stability of the Brin-Pesin asymptotic holonomy group for frame flows. Finally, we obtain local rigidity properties for uniform lattice actions on the ideal boundary of quaternionic and octonionic symmetric spaces.

math.DS↗
arXiv2021

DynACPD Embedding Algorithm for Prediction Tasks in Dynamic Networks

Classical network embeddings create a low dimensional representation of the learned relationships between features across nodes. Such embeddings are important for tasks such as link prediction and node classification. In the current paper, we consider low dimensional embeddings of dynamic networks, that is a family of time varying networks where there exist both temporal and spatial link relationships between nodes. We present novel embedding methods for a dynamic network based on higher order tensor decompositions for tensorial representations of the dynamic network. In one sense, our embeddings are analogous to spectral embedding methods for static networks. We provide a rationale for our algorithms via a mathematical analysis of some potential reasons for their effectiveness. Finally, we demonstrate the power and efficiency of our approach by comparing our algorithms' performance on the link prediction task against an array of current baseline methods across three distinct real-world dynamic networks.

cs.AI↗
arXiv2018

Homological norms on nonpositively curved manifolds

We relate the Gromov norm on homology classes to the harmonic norm on the dual cohomology and obtain double sided bounds in terms of the volume and other geometric quantities of the underlying manifold. Along the way, we provide comparisons to other related norms and quantities as well.

math.GT↗
arXiv2018

Maximal volume entropy rigidity for $\mathsf{RCD}^*(-(N-1),N)$ spaces

For $n$-dimensional Riemannian manifolds $M$ with Ricci curvature bounded below by $-(n-1)$, the volume entropy is bounded above by $n-1$. If $M$ is compact, it is known that the equality holds if and only if $M$ is hyperbolic. We extend this result to $\mathsf{RCD}^{\ast}(-(N-1),N)$ spaces. While the upper bound is straightforward, the rigidity case is quite involved due to the lack of a smooth structure in $\mathsf{RCD}^{\ast}$ spaces. As an application we obtain an almost rigidity result which partially recovers a result by Cheng-Rong-Xu for Riemannian manifolds.

math.DG↗
arXiv2018

Some remarks on the simplicial volume of nonpositively curved manifolds

We show that any closed manifold with a metric of nonpositive curvature that admits either a single point rank condition or a single point curvature condition has positive simplicial volume. We use this to provide a differential geometric proof of a conjecture of Gromov in dimension three.

math.GT↗
arXiv2018

The Circular Law for Random Matrices with Intra-row Dependence

We consider the problem of determining the limiting spectral distribution for random matrices whose row distributions are permitted to have limited dependence. We assume mild moment conditions and give an extension of the Mar\v{c}enko-Pastur theorem for this context. The main new feature here are geometric conditions on the distributions which allow us to extend the circular law to this setting.

math.PR↗
arXiv2017

Hyperbolic rank rigidity for manifolds of $\frac14$-pinched negative curvature

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac14]$, and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial converse to Hamenst\"{a}dt's hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

math.DG↗
arXiv2016

Vanishing simplicial volume for certain affine manifolds

We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. This provides some further evidence for the veracity of the Auslander Conjecture. Along the way, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups of $\pi_1$ to have vanishing simplicial volume. This answers a special case of a question due to L\"uck.

math.GT↗
arXiv2015

Harmonic and invariant measures on foliated spaces

We consider the family of harmonic measures on a lamination $\mathcal{L}$ of a compact space $X$ by locally symmetric spaces $L$ of noncompact type, i.e. $L\simeq \Gamma_L\backslash G/K$. We establish a natural bijection between these measures and the measures on an associated lamination foliated by $G$-orbits, $\hat{\mathcal{L}}$ which are right invariant under a minimal parabolic (Borel) subgroup $B < G$. In the special case when $G$ is split, these measures correspond to the measures that are invariant under both the Weyl chamber flow and the stable horospherical flows on a certain bundle over the associated Weyl chamber lamination. We also show that the measures on $\hat{\mathcal{L}}$ right invariant under two distinct minimal parabolics, and therefore all of $G$, are in bijective correspondence with the holonomy-invariant ones.

math.DS↗