In this short note, exploits of constructions of F-structures coupled with technology developed by Cheeger–Gromov and Paternain–Petean are seen to yield a procedure to compute minimal entropy, minimal volume, Yamabe invariant and to study collapsing with bounded sectional curvature on inequivalent smooth structures and inequivalent PL-structures within a fixed homeomorphism class. We compute these fundamental Riemannian invariants for every high-dimensional smooth manifold on the homeomorphism class of any smooth manifold that admits a Riemannian metric of zero sectional curvature. This includes all exotic and all fake tori of dimension greater than four. We observe that the minimal volume is not an invariant of the smooth structures, yet the Yamabe invariant does discern the standard smooth structure from all the others. We also observe that the fundamental group places no restriction on the vanishing of the minimal volume and collapse with bounded sectional curvature for high-dimensional manifolds.

Some examples of vanishing Yamabe invariant and minimal volume, and collapsing of inequivalent smoothings and PL-structures / Torres Ruiz, Rafael. - In: ANNALS OF GLOBAL ANALYSIS AND GEOMETRY. - ISSN 1572-9060. - 49:2(2016), pp. 177-194. [10.1007/s10455-015-9486-9]

Some examples of vanishing Yamabe invariant and minimal volume, and collapsing of inequivalent smoothings and PL-structures

Torres Ruiz, Rafael
2016-01-01

Abstract

In this short note, exploits of constructions of F-structures coupled with technology developed by Cheeger–Gromov and Paternain–Petean are seen to yield a procedure to compute minimal entropy, minimal volume, Yamabe invariant and to study collapsing with bounded sectional curvature on inequivalent smooth structures and inequivalent PL-structures within a fixed homeomorphism class. We compute these fundamental Riemannian invariants for every high-dimensional smooth manifold on the homeomorphism class of any smooth manifold that admits a Riemannian metric of zero sectional curvature. This includes all exotic and all fake tori of dimension greater than four. We observe that the minimal volume is not an invariant of the smooth structures, yet the Yamabe invariant does discern the standard smooth structure from all the others. We also observe that the fundamental group places no restriction on the vanishing of the minimal volume and collapse with bounded sectional curvature for high-dimensional manifolds.
2016
49
2
177
194
https://arxiv.org/abs/1502.03731
http://cdsads.u-strasbg.fr/abs/2015arXiv150203731T
Torres Ruiz, Rafael
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/33188
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