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

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 g₀, then ent(g) is minimized uniquely by g₀ 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).

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Published 1 January 2003