Abstract.
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 obtained by taking connected sums of copies of S 4, ℂP 2, \(\overline {\mathbb{C}P} ^2\),S 2×S 2and 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×S 2,ℂP 2\(\# \overline {\mathbb{C}P} ^2\) 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×S 2, then on trivial S 3-bundle over S 2 or the Wu-manifold SU(3)/SO(3).
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Oblatum 13-III-2002 & 12-VIII-2002¶Published online: 8 November 2002
G.P. Paternain was partially supported by CIMAT, Guanajuato, México.¶J. Petean is supported by grant 37558-E of CONACYT.
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Paternain, G., Petean, J. Minimal entropy and collapsing with curvature bounded from below. Invent. math. 151, 415–450 (2003). https://doi.org/10.1007/s00222-002-0262-7
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DOI: https://doi.org/10.1007/s00222-002-0262-7