Minimal entropy and collapsing with curvature bounded from below

• 成果类型: Article
• 署名作者: Paternain, GP; Petean, J
• 署名单位: University of Cambridge; CIMAT - Centro de Investigacion en Matematicas
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-002-0262-7
• 发表日期: 2003
• 页码: 415-450
• 关键词: topological-entropy; invariant; MANIFOLDS

摘要

We show that if a closed manifold M admits an F-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S-1 -action. As a corollary we obtain that the simplicial volume of a 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 in 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 closed 5-manifold. We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold of S-4, Cp-2 CP2, S-2 X S-2 obtained by taking connected sums of copies CP and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S-4, Cp-2 S-2 x S-2, Cp-2 CP2 or Cp-2 Cp-2. Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S-5, S-3 x S-2, the nontrivial S-3-bundle over S-2 or the Wu-manifold SU(3)/SO(3).