On entropies, $${\mathcal {F}}$$ F -structures, and scalar curvature of certain involutions

Abstract

In this short note, we analyze geometric properties of orbit spaces of certain involutions in dimensions four, five, and six. We consider constructions of F -structures on manifolds of dimension at least four that allows us to study minimal entropy, minimal volume, collapse with bounded curvature, and sign of the Yamabe invariant, and its vanishing, building on work of Paternain-Petean. The existence of Riemannian metrics of vanishing topological entropy on the orbit spaces is investigated as well.

A note on collapse, entropy, and vanishing of the Yamabe invariant of symplectic 4-manifolds

“…This has been previously observed for Kähler surfaces by LeBrun [L99], and on the homeomorphism type of the K3 surface by the second author of this note [To14].…”

“…No symplectic 4-manifold of Kodaira dimension one realizes its Yamabe invariant, and neither do several symplectic 4-manifolds of Kodaira dimension zero. This has been previously observed for Kähler surfaces by LeBrun [L99], and on the homeomorphism type of the K3 surface by the second author of this note [To14].…”

“…Hitchin's result [Be87, 6.37 Theorem] states that a Ricci-flat oriented 4-manifold has either zero sectional curvature, is the K3 surface, the Enriques surface, or a quotient of the later by a free antiholomorphic involution. Since the holomorphic Kodaira dimension coincides with the symplectic one, this immediately implies that no symplectic 4-manifold of Kodaira dimension one can realize a vanishing Yamabe invariant, and restricts those of Kodaira dimension zero that do realize it (see [L99,To14]).…”