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 admits a geometric decomposition in the sense of Thurston and does not have geometric pieces modelled on hyperbolic four-space , the complex hyperbolic plane or the product of two hyperbolic planes then admits an –structure. It follows that has zero minimal entropy and collapses with curvature bounded from below. We then analyse whether or not admits a metric whose topological entropy coincides with the minimal entropy of and provide new examples of manifolds for which the minimal entropy problem cannot be solved.
Citation
Pablo Suárez-Serrato. "Minimal entropy and geometric decompositions in dimension four." Algebr. Geom. Topol. 9 (1) 365 - 395, 2009. https://doi.org/10.2140/agt.2009.9.365
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