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Entropies, Volumes, and Einstein Metrics

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Global Differential Geometry

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 17))

Abstract

We survey the definitions and some important properties of several asymptotic invariants of smooth manifolds, and discuss some open questions related to them. We prove that the (non-)vanishing of the minimal volume is a differentiable property, which is not invariant under homeomorphisms. We also formulate an obstruction to the existence of Einstein metrics on four-manifolds involving the volume entropy. This generalizes both the Gromov–Hitchin–Thorpe inequality proved in [Kotschick, On the Gromov–Hitchin–Thorpe inequality, C. R. Acad. Sci. Paris 326 (1998), 727–731], and Sambusetti’s obstruction [Sambusetti, An obstruction to the existence of Einstein metrics on 4-manifolds, Math. Ann. 311 (1998), 533–547].

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Acknowledgements

This work is part of the project Asymptotic invariants of manifolds, supported by the Schwerpunktprogramm Globale Differentialgeometrie of the Deutsche Forschungsgemeinschaft. © D. Kotschick 2004–2010

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Authors and Affiliations

  1. Mathematisches Institut, LMU München, Theresienstr. 39, 80333, München, Germany

    D. Kotschick

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  1. D. Kotschick
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Correspondence to D. Kotschick .

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Editors and Affiliations

  1. Inst. Mathematik, Universität Potsdam, Potsdam, Brandenburg, Germany

    Christian Bär

  2. , Mathematisches Institut, Universität Münster, Einsteinstraße 62, Münster, 48149, Germany

    Joachim Lohkamp

  3. , Fakultät für Mathematik und Informatik, Universität Leipzig, Johannisgasse 26, Leipzig, 04103, Sachsen, Germany

    Matthias Schwarz

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Kotschick, D. (2012). Entropies, Volumes, and Einstein Metrics. In: Bär, C., Lohkamp, J., Schwarz, M. (eds) Global Differential Geometry. Springer Proceedings in Mathematics, vol 17. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22842-1_2

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