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2009 Minimal entropy and geometric decompositions in dimension four
Pablo Suárez-Serrato
Algebr. Geom. Topol. 9(1): 365-395 (2009). DOI: 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 4, the complex hyperbolic plane 2 or the product of two hyperbolic planes 2×2 then M admits an –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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Pablo Suárez-Serrato. "Minimal entropy and geometric decompositions in dimension four." Algebr. Geom. Topol. 9 (1) 365 - 395, 2009.


Received: 21 April 2008; Revised: 5 February 2009; Accepted: 5 February 2009; Published: 2009
First available in Project Euclid: 20 December 2017

zbMATH: 1160.37321
MathSciNet: MR2482083
Digital Object Identifier: 10.2140/agt.2009.9.365

Primary: 37B40, 57M50
Secondary: 22F30, 53D25

Rights: Copyright © 2009 Mathematical Sciences Publishers


Vol.9 • No. 1 • 2009
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