Paper · Feb 1, 2016

Lower semicontinuity of mass under convergence and Huisken's isoperimetric mass

Jeffrey L. Jauregui, Dan A. Lee

math.DGWider flowsarXiv:1602.00732
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As on arXiv

Abstract

Given a sequence of asymptotically flat 3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed Cheeger–Gromov sense to an asymptotically flat limit space, we show that the total mass of the limit is bounded above by the liminf of the total masses of the sequence. In other words, total mass is lower semicontinuous under such convergence. In order to prove this, we use Huisken's isoperimetric mass concept, together with a modified weak mean curvature flow argument. We include a brief discussion of Huisken's work before explaining our extension of that work. The results are all specific to three dimensions.

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Submitted
Feb 1, 2016
Subjects
math.DG, gr-qc
Journal
Journal fuer die reine und angewandte Mathematik (Crelle's Journal), Vol. 2019, No. 756, pg. 227-257

Comments: 30 pages, 5 figures

Mean curvature flow3-manifoldsScalar curvatureBoundaries & non-compactGeometric inequalities

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Isoperimetric Riemannian Penrose Inequality

    Luca Benatti, Mattia Fogagnolo and Lorenzo MazzieriWider flows

    Topics Geometric inequalities, Scalar curvature, Boundaries & non-compact · Shared terms “Isoperimetric mass”, “Huisken Isoperimetric”, “nonnegative scalar”

  2. Monotone Quantities for -Harmonic functions and the Sharp -Penrose inequality

    Liam Mazurowski and Xuan YaoWider flows

    Topics Geometric inequalities, 3-manifolds, Scalar curvature · Shared terms “minimal boundary”, “asymptotically flat”

  3. Minkowski inequality in Cartan-Hadamard manifolds

    Mohammad Ghomi and Joel SpruckWider flows

    Topics Geometric inequalities, 3-manifolds · Shared terms “isoperimetric”

  4. Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature

    Gioacchino Antonelli, Mattia Fogagnolo, Stefano Nardulli et al.Wider flows

    Topics Geometric inequalities, Scalar curvature · Shared terms “nonnegative scalar”, “isoperimetric”

  5. Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension

    Yuguang Shi and Jintian ZhuWider flows

    Topics 3-manifolds, Boundaries & non-compact · Shared terms “total mass”, “asymptotically flat”

  6. Pseudolocality theorems of Ricci flows on incomplete manifolds

    Liang Cheng

    Topics Geometric inequalities, Scalar curvature, Boundaries & non-compact · Shared terms “isoperimetric”, “nonnegative scalar”

  7. Capacity inequalities and rigidity of cornered/conical manifolds

    C. Tiarlos CruzWider flows

    Topics Geometric inequalities, Boundaries & non-compact · Shared terms “asymptotically flat”

  8. The Yamabe flow on asymptotically flat manifolds

    Eric Chen and Yi WangWider flows

    Topics Scalar curvature, Boundaries & non-compact · Shared terms “asymptotically flat”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.