Communications in Mathematical Physics

, Volume 272, Issue 1, pp 119–138 | Cite as

Generalized Inverse Mean Curvature Flows in Spacetime

  • Hubert Bray
  • Sean Hayward
  • Marc Mars
  • Walter Simon


Motivated by the conjectured Penrose inequality and by the work of Hawking, Geroch, Huisken and Ilmanen in the null and the Riemannian case, we examine necessary conditions on flows of two-surfaces in spacetime under which the Hawking quasilocal mass is monotone. We focus on a subclass of such flows which we call uniformly expanding, which can be considered for null as well as for spacelike directions. In the null case, local existence of the flow is guaranteed. In the spacelike case, the uniformly expanding condition leaves a 1-parameter freedom, but for the whole family, the embedding functions satisfy a forward-backward parabolic system for which local existence does not hold in general. Nevertheless, we have obtained a generalization of the weak (distributional) formulation of this class of flows, generalizing the corresponding step of Huisken and Ilmanen’s proof of the Riemannian Penrose inequality.


Curvature Flow Generalize Inverse Local Existence Curvature Vector Curvature Condition 
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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Hubert Bray
    • 1
  • Sean Hayward
    • 2
    • 3
  • Marc Mars
    • 4
  • Walter Simon
    • 4
  1. 1.Mathematics DepartmentDuke UniversityDurhamUSA
  2. 2.Center for AstrophysicsShanghai Normal UniversityShanghaiChina
  3. 3.Center for Mathematical PhysicsEast China University of Science and TechnologyShanghaiChina
  4. 4.Facultad de CienciasUniversidad de SalamancaSalamancaSpain

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