Minimal entropy and geometric decompositions in dimension four

Abstract

We show vanishing results about the infimum of the topological entropy of the geodesic flow of homogeneous smooth four-manifolds. We prove that any closed oriented geometric four-manifold has zero minimal entropy if and only if it has zero simplicial volume. We also show that if a four-manifold $M$ admits a geometric decomposition in the sense of Thurston and does not have geometric pieces modelled on hyperbolic four-space ${ℍ}^4$, the complex hyperbolic plane ${ℍ}_{ℂ}^2$ or the product of two hyperbolic planes ${ℍ}^2\times {ℍ}^2$ then $M$ admits an $\mathcal{ℱ}$–structure. It follows that $M$ has zero minimal entropy and collapses with curvature bounded from below. We then analyse whether or not $M$ admits a metric whose topological entropy coincides with the minimal entropy of $M$ and provide new examples of manifolds for which the minimal entropy problem cannot be solved.