2000/11/15 by Gabriel Paternain, Paternain, Gabriel, Jimmy Petean +1
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #math.DG #math.DS
paper · pdf · doi:10.48550/arxiv.math/0011104
arxiv created 2000/11/15 · arxiv updated 2009/11/30
We show that if a closed manifold M admits an F-structure (possibly of rank 0) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial circle action. As a corollary we obtain that the simplicial volume of a colsed manifold admitting an F-structure is zero. We also show that if M admits an F-structure then it collapses with curvature bounded from below. This is turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative. We show that F-structures of rank zero appear rather frequently:every compact complex elliptic surface admits one as well as any simply connected 5-manifold. We use these results to study the minimal entropy problem. We show the following two theorems: suppose M is obtained by taking connected sums of copies of CP2 (with any orintation), S2 × S2 and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a metric with zero topological entropy if and only if M is diffeomorphic to S4, CP2, S2 × S2, CP2#CP2 or CP2#(-CP2). Finally, suppose that M is a closed simply connected 5-manifold. Than M has zero minimal entropy. Moreover, M admits a metric with zero topological entropy if and only if M is diffeomorphic to S5, S3 × S2, the non-trivial S3-bundle over S2 or the Wu manifold SU(3)/SO(3).