Chris Connell
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Chris Connell has a total of 28 co-authors. Here are the top co-authors by number of shared publications:
- Pablo Suárez-Serrato
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
4 shared publications - Ralf Spatzier
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
4 shared publications - Shi Wang
Indiana University, Bloomington, USA
3 shared publications - Thang Nguyen
Univ Michigan, Dept Math, Ann Arbor, MI 48109 USA
3 shared publications - Guofang Wei
Department of Mathematics, University of California, Santa Barbara, CA 93106, USA
2 shared publications - Jesús Núñez-Zimbrón
Universidad Nacional Autónoma de México
2 shared publications
Latest Publications
- Open Access
- 28 May 2026
- Published by Springer Nature in Mathematische Annalen
- Vol. 395 (3)
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 $$0 \le \kappa \le 1.$$ 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.- 28 May 2026
- Published by Springer Nature in Mathematische Annalen
- Vol. 395 (3)
Open AccessRigidity 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 $$0 \le \kappa \le 1.$$ 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. - Open Access
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.Open AccessNonpositively 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. - Open AccessBoundary actions of lattices and $C^0$ local semi-rigidityWe 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.Open AccessBoundary actions of lattices and $C^0$ local semi-rigidityWe 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.
- Open Access
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.
Open AccessHomological 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.
- Open Access
On Higher Graph Manifolds
In this short note we introduce higher graph manifolds and use a version of the barycenter technique to characterize when they undergo volume collapse. In the case when the pure pieces are hyperbolic, we compute the exact value of the minimal volume. We verify the coarse Baum--Connes conjecture for these manifolds and show that they do not admit positive scalar curvature metrics. In the case without any pure pieces, we show the Yamabe invariant vanishes.
Open AccessOn Higher Graph Manifolds
In this short note we introduce higher graph manifolds and use a version of the barycenter technique to characterize when they undergo volume collapse. In the case when the pure pieces are hyperbolic, we compute the exact value of the minimal volume. We verify the coarse Baum--Connes conjecture for these manifolds and show that they do not admit positive scalar curvature metrics. In the case without any pure pieces, we show the Yamabe invariant vanishes.
- Open Access
- 9 December 2021
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 42 (2) , 614-664
Carnot metrics, dynamics and local rigidity
This paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and Mostow, Métivier, 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 Carnot–Carathéodory 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.
- 9 December 2021
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 42 (2) , 614-664
Open AccessCarnot metrics, dynamics and local rigidity
This paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and Mostow, Métivier, 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 Carnot–Carathéodory 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.
- 13 June 2021
- Published by Wiley in Journal of the London Mathematical Society
- Vol. 104 (4) , 1615-1681
Maximal volume entropy rigidity for RCD∗(−(N−1),N) spaces
- 13 June 2021
- Published by Wiley in Journal of the London Mathematical Society
- Vol. 104 (4) , 1615-1681
Maximal volume entropy rigidity for RCD∗(−(N−1),N) spaces
- Open AccessMaximal volume entropy rigidity for $\mathsf{RCD}^*(-(N-1),N)$ spacesFor $n$-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.Open AccessMaximal volume entropy rigidity for $\mathsf{RCD}^*(-(N-1),N)$ spacesFor $n$-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.
- 13 April 2020
- Published by Springer Nature in Mathematische Annalen
- Vol. 377 (3-4) , 969-987
Some remarks on the simplicial volume of nonpositively curved manifolds
- 13 April 2020
- Published by Springer Nature in Mathematische Annalen
- Vol. 377 (3-4) , 969-987
Some remarks on the simplicial volume of nonpositively curved manifolds
- Open Access
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.
Open AccessSome 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.