Minimal entropy and geometric decompositions in dimension four

Suárez-Serrato, Pablo

doi:10.2140/agt.2009.9.365

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Minimal entropy and geometric decompositions in dimension four

Pablo Suárez-Serrato

Algebraic & Geometric Topology 9 (2009) 365–395

DOI: 10.2140/agt.2009.9.365

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 $ℱ$ –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.

Keywords

minimal entropy, geodesic flows, geometric structures

Mathematical Subject Classification 2000

Primary: 37B40, 57M50

Secondary: 22F30, 53D25

References

Publication

Received: 21 April 2008

Revised: 5 February 2009

Accepted: 5 February 2009

Published: 25 February 2009

Authors

Pablo Suárez-Serrato
CIMAT Jalisco S/N Col. Valenciana CP: 36240 Guanajuato, Gto México.

Volume 9, issue 1 (2009)