Minimal entropy and collapsing with curvature bounded from below

By Gabriel P. Paternain, J. Petean. 2000

86 citations recorded by the source corpus.

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 S1-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 S4, ℂP2, $$\overline {\mathbb{C}P} ^2$$,S2×S2and 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 S4,ℂP2,S2×S2,ℂP2$$\# \overline {\mathbb{C}P} ^2$$ or ℂP2# ℂP2. 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 S5,S3×S2, then on trivial S3-bundle over S2 or the Wu-manifold SU(3)/SO(3).

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f42c6279e189e62cff1bd725f52f1b3cef2e18f1

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In Applied Mathematics

W. Ziller

In Geometry and Topology

Citations in the corpus

Examples of Riemannian Manifolds with non-negative sectional curvature (2007, background): “In [65] it was shown that an elliptic 5-manifold is diffeomorphic to one of the known examples with non-negative curvature, i.”
Riemannian Manifolds with Positive Sectional Curvature (2012, background): “tion. If n = 4, rational homotopy theory implies that M, if compact and simply connected, is diﬀeomorphic to one of the known examples of non-negative curvature, i.e. S4,CP2,S2 × S2 or CP2# ± CP2. In [PP] it was shown that a compact simply connected elliptic 5-manifold is diﬀeomorphic to one of the ”