Minimal entropy and geometric decompositions in dimension four

Details Minimal entropy and geometric decompositions in dimension four Minimal entropy and geometric decompositions in dimension four

2006-01-31

arxiv.org/abs/math/0601759v3

Algebraic & Geometric Topology 9 (2009) 365\^a?"395

Mathematics

Geometric Topology

38 pages, corrected flat case; Theorem A is now in terms of F-structures, improved exposition; new final section collects all

Scientific paper

10.2140/agt.2009.9.365

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, the complex hyperbolic plane or the product of two hyperbolic planes, then M admits an 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.

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