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The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality

Huisken, Gerhard, Ilmanen, Tom·2001·Journal of Differential Geometry
DOI 10.4310/jdg/1090349447
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Abstract

Let M be an asymptotically flat 3-manifold of nonnegative scalar curvature. The Riemannian Penrose Inequality states that the area of an outermost minimal surface N in M is bounded by the ADM mass m according to the formula |N| ≤ 16πm2. We develop a theory of weak solutions of the inverse mean curvature flow, and employ it to prove this inequality for each connected component of N using Geroch's monotonicity formula for the ADM mass. Our method also proves positivity of Bartnik's gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry.

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  • The classical Penrose inequality bounds the total mass M of a D ≥ 4 asymptotically flat or AdS spacetime with a marginally trapped surface σ below by the area A[σ] as in Eq. (7.11).

    “Loosely speaking, assuming there are no naked singularities and collapsing matter settles to a Kerr black hole, then the total mass M for a D ≥ 4-dimensional asymptotically flat [305][306][307][308] or AdS spacetime [309,310] with a marginally trapped surface σ is bounded below by the area A[σ]”

    From Three-Dimensional Quantum Black Holes: A Primer (2024) · cites this paper as [305]

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Research works printed but not linked: 26

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  2. R. Bartnik, Energy in General Relativity, Tsing Hua lectures on Geometry and Analysis (Hsinchu, 1990-1991), Internat. Press, Cambridge, MA (1997) 5-27.
  3. H. Bray, The Penrose Inequality in General Relativity and Volume Comparison Theorems Involving Scalar Curvature, thesis, Stanford University, 1997.
  4. D. Christodoulou & S. Klainerman, The Global Nonlinear Stability of Minkowski Space, Princeton Math. Series, 41, Princ. Univ. Press, Princeton, 1993.
  5. P.T. Chruściel, Thesis, Institute of Mathematics, Polish Academy of Sciences, 1985.
  6. P.T. Chruściel, On Kijowski's Unconstrained Degrees of Freedom of the Gravi- tational Field, Proc. of the IV Marcel Grossman Meeting on General Relativity (Rome, 1985), North Holland, New York (1986) 1643-1649.
  7. G.W. Gibbons, Thesis, Cambridge University, 1973.
  8. D. Gilbarg & N. Trudinger, Elliptic Partial Differential Equations of Second Or- der, 2nd ed., Springer, New York, 1983.
  9. M. Gromov & H.B. Lawson, The classification of simply connected manifolds of positive scalar curvature, Annals of Math. 111 (1980) 423-434.
  10. S.W. Hawking & R. Penrose, The singularities of gravitational collapse and cos- mology, Proc. Roy. Soc. A 314 (1970) 529-548.
  11. M. Heidusch, Higher Regularity of Inverse Mean Curvature Flow, preprint, Tü- bingen, 2001.
  12. J. Hempel, 3-Manifolds, Princ. Univ. Press, Princeton, 1976.
  13. G. Huisken & T. Ilmanen, A Note on Inverse Mean Curvature Flow, Proc. of the Workshop on Nonlinear Part. Diff. Equ. (Saitama Univ., Sept. 1997), Saitama Univ., 1997.
  14. J. Jezierski & J. Kijowski, A Simple Argument for Positivity of Gravitational Energy, Proceedings of the XV International Conference on Differential Geometric Methods in Theoretical Physics (Clausthal, 1986), World Sci. Pub., Teaneck NJ (1987) 187-194.
  15. J. Kijowski, Unconstrained Degrees of Freedom of Gravitational Field and the Positivity of Gravitational Energy, Gravitation, Geometry and Relativistic Physics (Aussois, 1984), Lect. Notes in Phys., 212, Springer, Berlin (1984) 40-50.
  16. J. Kijowski, On Positivity of Gravitational Energy, Proc. of the IV Marcel Gross- man Meeting on General Relativity (Rome, 1985), ed.
  17. F.-H. Lin, Regularity for a Class of Parametric Obstacle Problems, thesis, Univ. Minnesota, 1985.
  18. E. Pasch, Numerische Verfahren zur Berechnung von Krümmungsflüssen, Ph.D. thesis, Tübingen Univ., 1998.
  19. R. Penrose, Some Unsolved Problems in Classical General Relativity, Seminar on Differential Geometry, ed. S.-T. Yau, Annals of Math. Stud., 102, Princ. Univ. Press, Princeton (1982) 631-668.
  20. R. Schoen & S.-T. Yau, Incompressible Minimal Surfaces, Three-Dimensional Manifolds with Nonnegative Scalar Curvature, and the Positive Mass Conjecture in General Relativity, Proc. Nat. Acad. Sci. 75 (1978) 45-76.
  21. R. Schoen & S.-T. Yau, Complete Manifolds with Nonnegative Scalar Curvature and the Positive Action Conjecture in General Relativity, Proc. Nat. Acad. Sci. 75 (1978) 45-76.
  22. R. Schoen & S.-T. Yau, The Structure of Manifolds with Positive Scalar Curva- ture, Directions in Partial Differential Equations (Madison, 1985), Publ. Math. Res. Center Univ. Wisconsin, 54, Academic Press, Boston (1987) 235-242.
  23. L.M. Simon, Lectures on Geometric Measure Theory, Proc. of the Centre for Math. Analysis, 3, Austr. Nat. Univ., 1983.
  24. R.M. Wald, General Relativity, Univ. Chicago Press, Chicago, 1984.
  25. T. Willmore, Total Curvature in Riemannian Geometry, Wiley, New York, 1982.
  26. J.W. York, Jr., Kinematics and Dynamics of General Relativity, Sources of Grav- itational Radiation, ed. L. Smarr, Camb. Univ. Press, 1979, 83-126.

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Metadata

Authors
Huisken, G., & Ilmanen, T.
Venue
Journal of Differential Geometry
Publisher
International Press of Boston
Published
2001-11-01
Volume
59
Issue
3
Pages
353-437
ISSN
0022-040X
DOI
10.4310/jdg/1090349447
Cited by
463
References
1

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