Paper · Oct 14, 2020

A Minkowski inequality for the static Einstein-Maxwell space-time

B. Leandro, A. de Melo, H. Pina

math.DGWider flowsarXiv:2010.07187v7
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Abstract

In this paper we prove a Minkowski-like inequality for an asymptotically flat static Einstein-Maxwell (electrostatic) space-time using as approach the inverse mean curvature flow (IMCF). Moreover, we discuss the importance of this inequality in the understanding of the photon sphere.

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Submitted
Oct 14, 2020
Last revised
Mar 4, 2022 (v7)
Subjects
math.DG, math-ph
Mean curvature flowOther curvature flowsInverse mean curvature flowBoundaries & non-compactGeometric inequalities

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    Julian ScheuerWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Minkowski inequality”

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    Brian HarvieWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Boundaries & non-compact · Shared terms “Minkowski inequality”, “IMCF”

  3. A Minkowski type inequality in warped cylinders

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    Topics Geometric inequalities, Inverse mean curvature flow, Boundaries & non-compact · Shared terms “Minkowski”, “asymptotically flat”

  4. The Penrose inequality with charge for 2-convex initial data sets

    Tuan DolmenWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Boundaries & non-compact · Shared terms “Einstein Maxwell”, “asymptotically flat”

  5. Minkowski inequalities and constrained inverse curvature flows in warped spaces

    Julian ScheuerWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Minkowski inequality”

  6. New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms

    Jie WuWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Minkowski inequality”

  7. Capacity inequalities and rigidity of cornered/conical manifolds

    C. Tiarlos CruzWider flows

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

  8. Willmore inequality on hypersurfaces in hyperbolic space

    Yingxiang HuWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) 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. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.