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Generalized Inverse Mean Curvature Flows in Spacetime

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Abstract

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.

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Authors and Affiliations

  1. Mathematics Department, Duke University, Box 90320, Durham, NC, 27708, USA

    Hubert Bray

  2. Center for Astrophysics, Shanghai Normal University, 100 Guilin Road, Shanghai, 200234, China

    Sean Hayward

  3. Center for Mathematical Physics, East China University of Science and Technology, 130 Meilong Road, Shanghai, 200237, China

    Sean Hayward

  4. Facultad de Ciencias, Universidad de Salamanca, Plaza de la Merced s/n, 37008, Salamanca, Spain

    Marc Mars & Walter Simon

Authors
  1. Hubert Bray
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  2. Sean Hayward
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  3. Marc Mars
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  4. Walter Simon
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Correspondence to Marc Mars.

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Communicated by G.W. Gibbons

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Bray, H., Hayward, S., Mars, M. et al. Generalized Inverse Mean Curvature Flows in Spacetime. Commun. Math. Phys. 272, 119–138 (2007). https://doi.org/10.1007/s00220-007-0203-9

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  • DOI: https://doi.org/10.1007/s00220-007-0203-9

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