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# Communications in Analysis and Geometry

## Volume 11 (2003)

### Number 5

### Minimal Entropy Rigidity for Lattices in Products of Rank One Symmetric Spaces

Pages: 1001 – 1026

DOI: http://dx.doi.org/10.4310/CAG.2003.v11.n5.a7

#### Authors

#### Abstract

The *volume entropy h(g)* of a closed Riemannian *n*-manifold (*M, g*) is defined as

*h(g)* = lim_{ R→∞} 1/*R* log(Vol(*B(x,R*)))

where *B(x,R)* is the ball of radius *R* around a fixed point *x* in the universal cover *X*. (For noncompact *M*, see Section 6.2.) The number *h(g)* is independent of the choice of *x* , and equals the topological entropy of the geodesic flow on (*M, g*) when the curvature *K(g)* satisfies *K(g)* ≤ 0 (see [Ma]). Note that while the volume Vol(*M, g*) is not invariant under scaling the metric *g*, the *normalized entropy*

ent(*g*) = *h(g)*^{n} Vol(*M, g*)

is scale invariant.

Besson-Courtois-Gallot [BCG1] showed that, if *n ≥ 3* and *M* admits a negatively curved, locally symmetric metric *g*_{0}, then ent(*g*) is minimized *uniquely* by *g*_{0} in the space of all Riemannian metrics on *M*. This striking result, called *minimal entropy rigidity*, has a great number of corollaries, including solutions to long-standing problems on geodesic flows, asymptotic harmonicity, Gromov's minvol invariant, and a new proof of Mostow Rigidity in the rank one case (see [BCG2]).

Extending minimal entropy rigidity to all nonpositively curved, locally symmetric manifolds *M* has been a well-known open problem (see, e.g., [BCG2], Open Question 5). The case of closed manifolds locally (but not necessarily globally) isometric to products of negatively curved locally symmetric spaces of dimension at least 3 was announced in [BCG2] and later in [BCG3].

In this paper we prove minimal entropy rigidity in this case as well as in the more general setting of complete, finite volume manifolds. Although we haven't seen Besson-Courtois-Gallot's proof of this result, it is likely that our proof (in the compact case) overlaps with theirs. In particular, we apply the powerful method introduced in [BCG1, BCG2], with a few new twists (see below).