The Riemannian Penrose Inequality

International Mathematics Research Notices

Volumn , Issue 20, 1997, Pages 1044-1058

  Huisken, G ^a   Ilmanen, T ^b  

^a UNIVERSITY OF TÜBINGEN   (Germany)

^b MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 1842627896     PISSN: 10737928     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (57)

1. R. Arnowitt, S. Deser, and C. Misner, Coordinate invariance and energy expressions in general relativity, Phys. Rev. 122 (1961), 997-1006.
2. C. Barrabès and W. Israel, Thin shells in general relativity and cosmology: The lightlike limit, Phys. Rev. Lett. D 43 (1991), 1129-1142.
3. R. Bartnik, Existence of maximal surfaces in asymptotically flat spacetimes, Comm. Math. Phys. 94 (1984), 155-175.
4. _, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), 661-693.
5. _, Quasi-spherical metrics and prescribed scalar curvature, J. Differential Geom. 37 (1993), 31-71.
6. _, New definition of quasi-local mass, Phys. Rev. Lett. 62 (1989), 2346-2348.
7. H. Bray, The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature, Ph.D. thesis, Stanford University, 1997.
8. _, Isoperimetric techniques and the Penrose inequality in general relativity, in preparation, 1997.
9. D. Christodoulou, Examples of naked singularity formation in the gravitational collapse of a scalar field, Ann. of Math. 140 (1994), 607-653.
10. D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of Minkowski Space, Princeton Math. Ser. 41, Princeton Univ. Press, Princeton, 1993.
11. D. Christodoulou and S. T. Yau, "Some remarks on the quasi-local mass" in Mathematics and General Relativity (Santa Cruz, CA, 1986), J. Isenberg, ed., Contemp. Math. 71, Amer. Math. Soc., Providence, 1988, 9-14.
12. P. Chruściel, "Boundary conditions at spatial infinity from a Hamiltonian point of view" in Topological Properties and Global Structure of Space-Time (Erice, 1985), NATO Adv. Sci. Inst. Ser. B Phys. 138, Plenum, New York, 1986, 49-59.
13. L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Reg. Conf. Ser. in Math. 74, Amer. Math. Soc., Providence, 1990.
14. L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom. 33 (1991), 635-681.
15. H. Friedrich, Hyperbolic reductions for Einstein's equations, Classical Quantum Gravity 13 (1996), 1451-1469.
16. C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom. 32 (1990), 299-314.
17. R. Geroch, Energy extraction, Ann. New York Acad. Sci. 224 (1973), 108-117.
18. G. W. Gibbons, Collapsing shells and the isoperimetric inequality for black holes, Univ. of Camb. Preprint hep-th/9701049, preprint, 1997.
19. G. Gibbons, "The isoperimetric and Bogomolny inequalities for black holes" in Global Riemannian Geometry (Durham, 1983), T. Willmore and N. Hitchin, eds., Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1984, 194-202.
20. M. Gromov and H. B. Lawson, Spin and scalar curvature in the presence of a fundamental group, I, Ann. of Math. 111 (1980), 209-230.
21. _, The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. 111 (1980), 423-434.
22. _, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83-196.
23. R. Hardt and X. Zhou, An evolution problem for linear growth functionals, Comm. Partial Differential Equations 19 (1994), 1879-1907.
24. S. W. Hawking, Gravitational radiation in an expanding universe, J. Math. Phys. 9 (1968), 598-604.
25. _, Black holes in general relativity, Comm. Math. Phys. 25 (1972), 152-166.
26. S. W. Hawking and G. F. R. Ellis, The Large-Scale Structure of Space-Time, Cambridge Monogr. Math. Phys. 1, Cambridge Univ. Press, London, 1973.
27. S. W. Hawking and R. Penrose, The singularities of gravitational collapse and cosmology, Proc. Roy. Soc. London Ser. A 314 (1970), 529-548.
28. M. Herzlich, A Penrose-like inequality for the mass of Riemannian asymptotically flat manifolds, Prepublications du Centre de Math. Ecole Polytechnique #1146, 1996, to appear in Comm. Math. Phys.
29. M. Holder, Contracting spacelike hypersurfaces by their inverse mean curvature, Preprint SFB 382 Tübingen, No. 57, 1997.
30. G. Huisken and S. T. Yau, Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature, Invent. Math. 124 (1996), 281-311.
31. G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, in preparation.
32. T. Ilmanen, Elliptic Regularization and Partial Regularity for Motion by Mean Curvature, Amer. Math. Soc., Providence, 1994.
33. P. S. Jang, On the positive energy conjecture, J. Math. Phys. 17 (1976), 141-145.
34. _, On the positivity of energy in general relativity, J. Math. Phys. 19 (1978), 1152-1155.
35. _, Note on cosmic censorship, Phys. Rev. Lett. D 20 (1979), 834-838.
36. _, On positivity of mass for black hole space-times, Comm. Math. Phys. 69 (1979), 257-266.
37. P. S. Jang and R. M. Wald, The positive energy conjecture and the cosmic censor hypothesis, J. Math. Phys. 18 (1977), 41-44.
38. A. Lichnerowicz, Spineures Harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7-9.
39. S. Luckhaus, Solutions for the two-phase Stefan problem with the Gibbs-Thompson law for the melting temperature, European J. Appl. Math. 1 (1990), 101-111.
40. R. Penrose, Naked singularities, Ann. New York Acad. Sci. 224 (1973), 125-134.
41. O. Reula, Existence theorem for solutions of Witten's equations and nonnegativity of total mass, J. Math. Phys. 23 (1982), 810-814.
42. R. Schoen and 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.
43. _, Complete manifolds with nonnegative scalar curvature and the positive action conjecture in general relativity, Proc. Nat. Acad. Sci. 76 (1979), 1024-1025.
44. _, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), 45-76.
45. _, Positivity of the total mass of a general space-time, Phys. Rev. Lett. 43 (1979), 1457-1459.
46. _, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), 159-183.
47. _, Proof of the positive mass theorem, II, Comm. Math. Phys. 79 (1981), 231-260.
48. _, The energy and the linear momentum of space-times in general relativity, Comm. Math. Phys. 79 (1981), 47-51.
49. _, The existence of a black hole due to condensation of matter, Comm. Math. Phys. 90 (1983), 575-579.
50. _, "The structure of manifolds with positive scalar curvature" in Directions in Partial Differential Equations (Madison, 1985), Publ. Math. Res. Center Univ. Wisconsin 54, Academic Press, Boston, 1987, 235-242.
51. K. P. Tod, The hoop conjecture and the Gibbons-Penrose construction of trapped surfaces, Classical Quantum Gravity 9 (1992), 1581-1591.
52. N. S. Trudinger, A sharp inequality for subharmonic functions on two-dimensional manifolds, Math. Z. 133 (1973), 75-79.
53. J. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z. 205 (1990), 355-372.
54. A. Visintin, Nucleation and mean curvature flow, preprint.
55. B. White, On the topological type of minimal submanifolds, Topology 31 (1992), 445-448.
56. E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), 381-402.
57. J. W. York Jr., "Kinematics and dynamics of general relativity" in Sources of Gravitational Radiation, L. Smarr, ed., Cambridge University Press, Cambridge, 1979, p. 83.