arXiv subjects

Christopher Connell

Publications and source records attributed to Christopher Connell.

3 recordsLinked to original sources

arXiv2002

Some recent applications of the barycenter method in geometry

In this paper we describe some recent applications of the barycenter method in geometry. This method was first used by Duady-Earle and later greatly extended by Besson-Courtois-Gallot in their solution of a number of long-standing problems, in particular in their proof of entropy rigidity for closed, negatively curved locally symmetric manifolds. Since there are already a number of surveys describing this work, we will concentrate here only on advances that have occured after these surveys appeared. While most of this paper is a report on results appearing in other papers, some of the material here is new.

math.DG↗

arXiv2001

The Degree Theorem in higher rank

Let M be any closed, locally symmetric n-manifold (n>1) of nonpositive curvature. Assume that M has no locally Euclidean factors and no factors locally isometric to SL(3,R). Then for any closed Riemannian manifold N and any continuous map f:N -> M, we show there is a C^1 representative in the homotopy class of f with Jacobian bounded by a universal constant C depending only on n and the smallest Ricci curvatures of N and M. This implies that deg(f)<= C Vol(N)/Vol(M). For M negatively curved this was proved by Gromov. Two corollaries of our result are that Minvol(N)>0 whenever deg(f)<>0 and a simple proof of G. Prasad's result that lattices in semi-simple Lie groups are co-Hopfian. We prove a version of all results for finite volume manifolds as well.

math.DG↗

arXiv2001

Minimal entropy rigidity for lattices in products of rank one symmetric spaces

We prove minimal entropy rigidity for complete, finite volume manifolds locally isometric to a product of rank one symmetric spaces of dimension at least 3: the locally symmetric metric uniquely minimizes (normalized) entropy among all Riemannian metrics. The corresponding theorem is true for maps into these spaces as well.

math.DG↗