Communications in Analysis and Geometry
Volume 11 (2003)
Minimal Entropy Rigidity for Lattices in Products of Rank One Symmetric Spaces
Pages: 1001 – 1026
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 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).