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Four Lectures on Asymptotically Flat Riemannian Manifolds

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Einstein Equations: Physical and Mathematical Aspects of General Relativity (DOMOSCHOOL 2018)

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Abstract

After an introduction to the mathematical description of isolated gravitating systems, we present a proof of the positive mass theorem relying on the compactification trick that lies behind the argument proposed by Schoen and Yau in 2017 to handle the possible occurrence of high-dimensional singularities of minimizing cycles. From there, we describe some rigidity phenomena involving Riemannian manifolds of non-negative scalar curvature, and survey various recent results concerning the large-scale geometric structure of asymptotically flat spaces. In the last section, we outline the construction, due to Schoen and the author, of localized solutions of the Einstein constraints and discuss its implications both on the physical and the geometric side.

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Notes

  1. 1.

    For the sake of clarity, we shall always tacitly assume the word smooth to mean C ∞ and leave the straightforward modifications needed to handle the case of finite regularity, as encoded by functional spaces like C k, α or W k, p, to the reader. Such spaces will be relevant at a later stage and we will recall their definitions at due course.

  2. 2.

    We refer to compactness of currents with respect to the weak∗ topology, namely with respect to the canonical duality with spaces of differential forms.

  3. 3.

    By its very definition, considering the special case of ω 1 ∧… ∧ ω n−1 as a test form.

  4. 4.

    In fact, it is a standard result in geometric topology that any smooth map F : S h → S k has degree zero if h < k where S h (resp. S k) is a genus h (resp. k) surface.

  5. 5.

    A smooth, embedded, hypersurface is called area-minimizing if it is a local minimum for the area functional under smooth perturbations. In particular, an area-minimizing hypersurface is a stable, minimal hypersurface.

  6. 6.

    Recall from Definition 3.3 that we are assuming, for the sake of simplicity, to only deal with asymptotically flat manifolds with one end. More generally, one will have limr→+∞ Θ(r) ≥ 1 and thus Theorem 4.5 still holds true, with the very same proof.

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Author information

Authors and Affiliations

  1. IAS, School of Mathematics, Princeton, NJ, USA

    Alessandro Carlotto

  2. ETH, Department of Mathematics, Zürich, Switzerland

    Alessandro Carlotto

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  1. Alessandro Carlotto
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Correspondence to Alessandro Carlotto .

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Editors and Affiliations

  1. Department of Science and High Technology, University Insubrien, Como, Italy

    Sergio Cacciatori

  2. Mathematisches Institut, Humboldt-Universität zu Berlin, Berlin, Germany

    Batu Güneysu

  3. Department of Science and High Technology, University Insubrien, Como, Italy

    Stefano Pigola

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Carlotto, A. (2019). Four Lectures on Asymptotically Flat Riemannian Manifolds. In: Cacciatori, S., Güneysu, B., Pigola, S. (eds) Einstein Equations: Physical and Mathematical Aspects of General Relativity. DOMOSCHOOL 2018. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-18061-4_1

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