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Publications of Hubert Bray    :chronological  alphabetical  combined listing:

%% Papers Published   
@article{fds340288,
   Author = {Sormani, C and Bray, HL and Minicozzi, WP and Eichmair, M and Huang,
             L-H and Yau, S-T and Uhlenbeck, K and Kusner, R and Codá marques, F and Mese, C and Fraser, A},
   Title = {The Mathematics of Richard Schoen},
   Journal = {Notices of the American Mathematical Society},
   Volume = {65},
   Number = {11},
   Pages = {1-1},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2018},
   Month = {December},
   url = {http://dx.doi.org/10.1090/noti1749},
   Doi = {10.1090/noti1749},
   Key = {fds340288}
}

@article{fds330841,
   Author = {Bray, H and Roesch, H},
   Title = {Proof of a Null Geometry Penrose Conjecture},
   Journal = {Notices of the American Mathematical Society.},
   Volume = {65},
   Publisher = {American Mathematical Society},
   Year = {2018},
   Month = {February},
   Key = {fds330841}
}

@article{fds300017,
   Author = {Bray, HL and Jauregui, JL and Mars, M},
   Title = {Time Flat Surfaces and the Monotonicity of the Spacetime
             Hawking Mass II},
   Journal = {Annales Henri Poincaré},
   Volume = {17},
   Number = {6},
   Pages = {1457-1475},
   Publisher = {Springer Nature},
   Year = {2016},
   Month = {June},
   ISSN = {1424-0637},
   url = {http://arxiv.org/abs/1402.3287},
   Abstract = {© 2015, Springer Basel. In this sequel paper, we give a
             shorter, second proof of the monotonicity of the Hawking
             mass for time flat surfaces under spacelike uniformly area
             expanding flows in spacetimes that satisfy the dominant
             energy condition. We also include a third proof which builds
             on a known formula and describe a class of sufficient
             conditions of divergence type for the monotonicity of the
             Hawking mass. These flows of surfaces may have connections
             to the problem in general relativity of bounding the total
             mass of a spacetime from below by the quasi-local mass of
             spacelike 2-surfaces in the spacetime.},
   Doi = {10.1007/s00023-015-0420-2},
   Key = {fds300017}
}

@article{fds303060,
   Author = {Martinez-Medina, LA and Bray, HL and Matos, T},
   Title = {On wave dark matter in spiral and barred
             galaxies},
   Journal = {Journal of Cosmology and Astroparticle Physics},
   Volume = {2015},
   Number = {12},
   Pages = {025-025},
   Publisher = {IOP Publishing},
   Year = {2015},
   Month = {December},
   url = {http://arxiv.org/abs/1505.07154},
   Abstract = {© 2015 IOP Publishing Ltd and Sissa Medialab srl. We
             recover spiral and barred spiral patterns in disk galaxy
             simulations with a Wave Dark Matter (WDM) background (also
             known as Scalar Field Dark Matter (SFDM), Ultra-Light Axion
             (ULA) dark matter, and Bose-Einstein Condensate (BEC) dark
             matter). Here we show how the interaction between a baryonic
             disk and its Dark Matter Halo triggers the formation of
             spiral structures when the halo is allowed to have a
             triaxial shape and angular momentum. This is a more
             realistic picture within the WDM model since a non-spherical
             rotating halo seems to be more natural. By performing
             hydrodynamic simulations, along with earlier test particles
             simulations, we demonstrate another important way in which
             wave dark matter is consistent with observations. The common
             existence of bars in these simulations is particularly
             noteworthy. This may have consequences when trying to obtain
             information about the dark matter distribution in a galaxy,
             the mere presence of spiral arms or a bar usually indicates
             that baryonic matter dominates the central region and
             therefore observations, like rotation curves, may not tell
             us what the DM distribution is at the halo center. But here
             we show that spiral arms and bars can develop in DM
             dominated galaxies with a central density core without
             supposing its origin on mechanisms intrinsic to the baryonic
             matter.},
   Doi = {10.1088/1475-7516/2015/12/025},
   Key = {fds303060}
}

@article{fds287065,
   Author = {Bray, HL and Jauregui, JL},
   Title = {On curves with nonnegative torsion},
   Journal = {Archiv Der Mathematik},
   Volume = {104},
   Number = {6},
   Pages = {561-575},
   Publisher = {Springer Nature},
   Year = {2015},
   Month = {June},
   ISSN = {0003-889X},
   url = {http://www.springer.com/-/0/c8d239381b86496b96d95ff26f1061eb},
   Abstract = {© 2015, Springer Basel. We provide new results and new
             proofs of results about the torsion of curves in
             $${\mathbb{R}^3}$$ R3 . Let $${\gamma}$$γ be a smooth curve
             in $${\mathbb{R}^3}$$ R3 that is the graph over a simple
             closed curve in $${\mathbb{R}^2}$$ R2 with positive
             curvature. We give a new proof that if $${\gamma}$$γ has
             nonnegative (or nonpositive) torsion, then $${\gamma}$$γ
             has zero torsion and hence lies in a plane. Additionally, we
             prove the new result that a simple closed plane curve,
             without any assumption on its curvature, cannot be perturbed
             to a closed space curve of constant nonzero torsion. We also
             prove similar statements for curves in Lorentzian
             $${\mathbb{R}^{2,1}}$$ R2,1 which are related to important
             open questions about time flat surfaces in spacetimes and
             mass in general relativity.},
   Doi = {10.1007/s00013-015-0767-0},
   Key = {fds287065}
}

@article{fds300016,
   Author = {Bray, HL and Jauregui, JL},
   Title = {Time Flat Surfaces and the Monotonicity of the Spacetime
             Hawking Mass},
   Journal = {Communications in Mathematical Physics},
   Volume = {335},
   Number = {1},
   Pages = {285-307},
   Publisher = {Springer Nature},
   Year = {2015},
   Month = {January},
   ISSN = {0010-3616},
   url = {http://arxiv.org/abs/1310.8638},
   Abstract = {© 2014, Springer-Verlag Berlin Heidelberg. We identify a
             condition on spacelike 2-surfaces in a spacetime that is
             relevant to understanding the concept of mass in general
             relativity. We prove a formula for the variation of the
             spacetime Hawking mass under a uniformly area expanding flow
             and show that it is nonnegative for these so-called “time
             flat surfaces.” Such flows generalize inverse mean
             curvature flow, which was used by Huisken and Ilmanen to
             prove the Riemannian Penrose inequality for one black hole.
             A flow of time flat surfaces may have connections to the
             problem in general relativity of bounding the mass of a
             spacetime from below by the quasi-local mass of a spacelike
             2-surface contained therein.},
   Doi = {10.1007/s00220-014-2162-2},
   Key = {fds300016}
}

@article{fds287070,
   Author = {Bray, HL and Parry, AR},
   Title = {Modeling wave dark matter in dwarf spheroidal
             galaxies},
   Journal = {Journal of Physics: Conference Series},
   Volume = {615},
   Pages = {012001-012001},
   Publisher = {IOP Publishing},
   Year = {2015},
   ISSN = {1742-6588},
   url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000358144800001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
   Doi = {10.1088/1742-6596/615/1/012001},
   Key = {fds287070}
}

@article{fds287066,
   Author = {Bray, H and Goetz, AS},
   Title = {Wave Dark Matter and the Tully-Fisher Relation},
   Year = {2014},
   Month = {September},
   url = {http://arxiv.org/abs/1409.7347},
   Abstract = {Preprint},
   Key = {fds287066}
}

@article{fds287074,
   Author = {Bray, HL},
   Title = {On Dark Matter, Spiral Galaxies, and the Axioms of General
             Relativity},
   Journal = {Ams Contemporary Mathematics Volume},
   Volume = {599},
   Number = {Geometric Analysis, Mathematical Relativ},
   Publisher = {American Mathematical Society},
   Year = {2013},
   url = {http://www.math.duke.edu/~bray/darkmatter/darkmatter.html},
   Key = {fds287074}
}

@article{fds287075,
   Author = {Bray, HL and Jauregui, JL},
   Title = {A geometric theory of zero area singularities in general
             relativity},
   Journal = {Asian Journal of Mathematics},
   Volume = {17},
   Number = {3},
   Pages = {525-560},
   Publisher = {International Press of Boston},
   Year = {2013},
   ISSN = {1093-6106},
   url = {http://dx.doi.org/10.4310/AJM.2013.v17.n3.a6},
   Abstract = {The Schwarzschild spacetime metric of negative mass is
             well-known to contain a naked singularity. In a spacelike
             slice, this singularity of the metric is characterized by
             the property that nearby surfaces have arbitrarily small
             area. We develop a theory of such \zero area singularities"
             in Riemannian manifolds, generalizing far beyond the
             Schwarzschild case (for example, allowing the singularities
             to have nontrivial topology). We also dene the mass of such
             singularities. The main result of this paper is a lower
             bound on the ADM mass of an asymptotically at manifold of
             nonnegative scalar curvature in terms of the masses of its
             singularities, assuming a certain conjecture in conformal
             geometry. The proof relies on the Riemannian Penrose
             inequality [9]. Equality is attained in the inequality by
             the Schwarzschild metric of negative mass. An immediate
             corollary is a version of the positive mass theorem that
             allows for certain types of incomplete metrics. © 2013
             International Press.},
   Doi = {10.4310/AJM.2013.v17.n3.a6},
   Key = {fds287075}
}

@article{fds287064,
   Author = {Bray, H},
   Title = {On Wave Dark Matter, Shells in Elliptical Galaxies, and the
             Axioms of General Relativity},
   Year = {2012},
   Month = {December},
   url = {http://www.math.duke.edu/~bray/darkmatter/DMEG.pdf},
   Abstract = {Preprint},
   Key = {fds287064}
}

@article{fds287073,
   Author = {Bray, HL and Khuri, MA},
   Title = {P. D. E. 'S which imply the penrose conjecture},
   Journal = {Asian Journal of Mathematics},
   Volume = {15},
   Number = {4},
   Pages = {557-610},
   Publisher = {International Press of Boston},
   Year = {2011},
   Month = {January},
   ISSN = {1093-6106},
   url = {http://www.math.duke.edu/~bray/PE/euclid.ajm.1331583349.pdf},
   Abstract = {In this paper, we show how to reduce the Penrose conjecture
             to the known Riemannian Penrose inequality case whenever
             certain geometrically motivated systems of equations can be
             solved. Whether or not these special systems of equations
             have general existence theories is therefore an important
             open problem. The key tool in our method is the derivation
             of a new identity which we call the generalized Schoen-Yau
             identity, which is of independent interest. Using a
             generalized Jang equation, we use this identity to propose
             canonical embeddings of Cauchy data into corresponding
             static spacetimes. In addition, we discuss the Carrasco-Mars
             counterexample to the Penrose conjecture for generalized
             apparent horizons (added since the first version of this
             paper was posted on the arXiv) and instead conjecture the
             Penrose inequality for time-independent apparent horizons,
             which we define. © 2011 International Press.},
   Doi = {10.4310/AJM.2011.v15.n4.a5},
   Key = {fds287073}
}

@article{fds287068,
   Author = {Bray, H},
   Title = {On the Positive Mass, Penrose, and ZAS Inequalities in
             General Dimension},
   Booktitle = {Surveys in Geometric Analysis and Relativity in Honor of
             Richard Schoen’s 60th Birthday},
   Publisher = {Higher Education Press and International
             Press},
   Address = {Beijing and Boston},
   Editor = {Bray, H and Minicozzi, W},
   Year = {2011},
   url = {http://arxiv.org/abs/1101.2230},
   Key = {fds287068}
}

@article{fds287080,
   Author = {Bray, H and Brendle, S and Eichmair, M and Neves,
             A},
   Title = {Area-Minimizing Projective Planes in 3-Manifolds},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {63},
   Number = {9},
   Pages = {1237-1247},
   Publisher = {WILEY},
   Year = {2010},
   Month = {September},
   ISSN = {0010-3640},
   url = {http://dx.doi.org/10.1002/cpa.20319},
   Abstract = {Let (M, g) be a compact Riemannian manifold of dimension 3,
             and let F denote the collection of all embedded surfaces
             homeomorphic to R{double-struck}P{double-struck}2. We study
             the infimum of the areas of all surfaces in F . This
             quantity is related to the systole of .M; g/. It makes sense
             whenever F is nonempty. In this paper, we give an upper
             bound for this quantity in terms of the minimum of the
             scalar curvature of (M, g) Moreover, we show that equality
             holds if and only if (M, g) is isometric to
             R{double-struck}P{double-struck}3 up to scaling. The proof
             uses the formula for the second variation of area and
             Hamilton's Ricci flow. © 2010 Wiley Periodicals,
             Inc.},
   Doi = {10.1002/cpa.20319},
   Key = {fds287080}
}

@article{fds287081,
   Author = {Bray, HL and Khuri, MA},
   Title = {A jang equation approach to the penrose inequality},
   Journal = {Discrete and Continuous Dynamical Systems Series
             A},
   Volume = {27},
   Number = {2},
   Pages = {741-766},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2010},
   Month = {June},
   ISSN = {1078-0947},
   url = {http://dx.doi.org/10.3934/dcds.2010.27.741},
   Abstract = {We introduce a generalized version of the Jang equation,
             designed for the general case of the Penrose Inequality in
             the setting of an asymptotically flat space-like
             hypersurface of a spacetime satisfying the dominant energy
             condition. The appropriate existence and regularity results
             are established in the special case of spherically symmetric
             Cauchy data, and are applied to give a new proof of the
             general Penrose Inequality for these data sets. When
             appropriately coupled with an inverse mean curvature flow,
             analogous existence and regularity results for the
             associated system of equations in the nonspherical setting
             would yield a proof of the full Penrose Conjecture. Thus it
             remains as an important and challenging open problem to
             determine whether this system does indeed admit the desired
             solutions.},
   Doi = {10.3934/dcds.2010.27.741},
   Key = {fds287081}
}

@article{fds287076,
   Author = {Bray, H and Brendle, S and Neves, A},
   Title = {Rigidity of area-minimizing two-spheres in
             three-manifolds},
   Journal = {Communications in Analysis and Geometry},
   Volume = {18},
   Number = {4},
   Pages = {821-830},
   Publisher = {International Press of Boston},
   Year = {2010},
   Month = {January},
   ISSN = {1019-8385},
   url = {http://dx.doi.org/10.4310/CAG.2010.v18.n4.a6},
   Abstract = {We give a sharp upper bound for the area of a minimal
             two-sphere in a three-manifold (M,g) with positive scalar
             curvature. If equality holds, we show that the universal
             cover of (M,g) is isometric to a cylinder.},
   Doi = {10.4310/CAG.2010.v18.n4.a6},
   Key = {fds287076}
}

@article{fds287077,
   Author = {Bray, HL and Lee, DA},
   Title = {On the Riemannian Penrose inequality in dimensions less than
             eight},
   Journal = {Duke Mathematical Journal},
   Volume = {148},
   Number = {1},
   Pages = {81-106},
   Publisher = {Duke University Press},
   Year = {2009},
   Month = {May},
   ISSN = {0012-7094},
   url = {http://www.math.duke.edu/~bray/PE/euclid.dmj.1240432192.pdf},
   Abstract = {The positive mass theorem states that a complete
             asymptotically flat manifold of nonnegative scalar curvature
             has nonnegative mass and that equality is achieved only for
             the Euclidean metric. The Riemannian Penrose inequality
             provides a sharp lower bound for the mass when black holes
             are present. More precisely, this lower bound is given in
             terms of the area of an outermost minimal hypersurface, and
             equality is achieved only for Schwarzschild metrics. The
             Riemannian Penrose inequality was first proved in three
             dimensions in 1997 by G. Huisken and T. Ilmanen for the case
             of a single black hole (see [HI]). In 1999, Bray extended
             this result to the general case of multiple black holes
             using a different technique (see [Br]). In this article, we
             extend the technique of [Br] to dimensions less than eight.
             Part of the argument is contained in a companion article by
             Lee [L]. The equality case of the theorem requires the added
             assumption that the manifold be spin. 2009 © Duke
             University Press.},
   Doi = {10.1215/00127094-2009-020},
   Key = {fds287077}
}

@article{fds287083,
   Author = {Bray, H and Miao, P},
   Title = {On the capacity of surfaces in manifolds with nonnegative
             scalar curvature},
   Journal = {Inventiones Mathematicae},
   Volume = {172},
   Number = {3},
   Pages = {459-475},
   Publisher = {Springer Nature},
   Year = {2008},
   Month = {June},
   ISSN = {0020-9910},
   url = {http://dx.doi.org/10.1007/s00222-007-0102-x},
   Abstract = {Given a surface in an asymptotically flat 3-manifold with
             nonnegative scalar curvature, we derive an upper bound for
             the capacity of the surface in terms of the area of the
             surface and the Willmore functional of the surface. The
             capacity of a surface is defined to be the energy of the
             harmonic function which equals 0 on the surface and goes to
             1 at ∞. Even in the special case of ℝ3, this is a new
             estimate. More generally, equality holds precisely for a
             spherically symmetric sphere in a spatial Schwarzschild
             3-manifold. As applications, we obtain inequalities relating
             the capacity of the surface to the Hawking mass of the
             surface and the total mass of the asymptotically flat
             manifold. © 2008 Springer-Verlag.},
   Doi = {10.1007/s00222-007-0102-x},
   Key = {fds287083}
}

@article{fds287084,
   Author = {Bray, H and Hayward, S and Mars, M and Simon, W},
   Title = {Generalized inverse mean curvature flows in
             spacetime},
   Journal = {Communications in Mathematical Physics},
   Volume = {272},
   Number = {1},
   Pages = {119-138},
   Publisher = {Springer Nature},
   Year = {2007},
   Month = {May},
   ISSN = {0010-3616},
   url = {http://dx.doi.org/10.1007/s00220-007-0203-9},
   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. ©
             Springer-Verlag 2007.},
   Doi = {10.1007/s00220-007-0203-9},
   Key = {fds287084}
}

@article{fds303538,
   Author = {Bray, HL},
   Title = {A family of quasi-local mass functionals with monotone
             flows},
   Pages = {323-329},
   Publisher = {World Scientific},
   Editor = {JC Zambrini},
   Year = {2006},
   Month = {January},
   ISBN = {981256201X},
   url = {http://dx.doi.org/10.1142/9789812704016_0030},
   Abstract = {© 2005 by World Scientific Publishing Co. Pte. Ltd. All
             rights reserved. We define a one parameter family of
             quasi-local mass functionals mc(Σ), 0 ≤ c ≤ ∞, which
             are nondecreasing on surfaces in 3-manifolds with
             nonnegative scalar curvature with respect to a one parameter
             family of flows. In the case that c = 0, m0(Σ) equals the
             Hawking mass of Σ2and the corresponding flow is inverse
             mean curvature flow. Then, following the arguments of Geroch
             [8], Jang and Wald [12], and Huisken and Ilmanen [9], we
             note that the generalization of their results for inverse
             mean curvature flow would imply that if mADMis the total
             mass of the complete, asymptotically flat 3-manifold with
             nonnegative scalar curvature, then mADM≥ mc(Σ) for all
             nonnegative c and all connected surfaces Σ which are not
             enclosed by surfaces with less area.},
   Doi = {10.1142/9789812704016_0030},
   Key = {fds303538}
}

@article{fds51387,
   Author = {H.L. Bray},
   Title = {The Positve Energy Theorem and Other Inequalities in
             GR},
   Booktitle = {The Encyclopedia of Mathematical Physics},
   Year = {2005},
   Key = {fds51387}
}

@article{fds287061,
   Author = {Bray, H},
   Title = {The Positve Energy Theorem and Other Inequalities},
   Booktitle = {The Encyclopedia of Mathematical Physics},
   Year = {2005},
   Key = {fds287061}
}

@article{MR2052359,
   Author = {Bray, HL and Neves, A},
   Title = {Classification of Prime 3-Manifolds with Yamabe Invariant
             Greater than RP^3},
   Journal = {Annals of Mathematics},
   Volume = {159},
   Number = {1},
   Pages = {407-424},
   Publisher = {Annals of Mathematics, Princeton U},
   Year = {2004},
   Month = {January},
   url = {http://annals.math.princeton.edu/2004/159-1/p09},
   Abstract = {In this paper we compute the σ-invariants (sometimes also
             called the smooth Yamabe invariants) of RP3 and RP2×S1
             (which are equal) and show that the only prime 3-manifolds
             with larger σ-invariants are S3, S2×S1, and S2×~S1 (the
             nonorientable S2 bundle over S1). More generally, we show
             that any 3-manifold with σ-invariant greater than RP3 is
             either S3, a connect sum with an S2 bundle over S1, or has
             more than one nonorientable prime component. A corollary is
             the Poincaré conjecture for 3-manifolds with σ-invariant
             greater than RP3. Surprisingly these results follow from the
             same inverse mean curvature flow techniques which were used
             by Huisken and Ilmanen in [7] to prove the Riemannian
             Penrose Inequality for a black hole in a spacetime. Richard
             Schoen made the observation [18] that since the constant
             curvature metric (which is extremal for the Yamabe problem)
             on RP3 is in the same conformal class as the Schwarzschild
             metric (which is extremal for the Penrose inequality) on RP3
             minus a point, there might be a connection between the two
             problems. The authors found a strong connection via inverse
             mean curvature flow.},
   Doi = {10.4007/annals.2004.159.407},
   Key = {MR2052359}
}

@article{fds287060,
   Author = {Bray, H and Chrusciel, PT},
   Title = {The Penrose Inequality},
   Booktitle = {The Einstein Equations and the Large Scale Behavior of
             Gravitational Fields (50 Years of the Cauchy Problem in
             General Relativity)},
   Publisher = {Birkhauser},
   Editor = {Chrusciel, PT and Friedrich, HF},
   Year = {2004},
   url = {arxiv:gr-qc/0312047v2},
   Key = {fds287060}
}

@article{fds43695,
   Author = {H.L. Bray},
   Title = {A Family of Quasi-local Mass Functionals with Monotone
             Flows},
   Booktitle = {Proceedings of the 14th International Congress on
             Mathematical Physics, Lisbon, Portugal, 2003},
   Editor = {Jean-Claude Zambrini},
   Year = {2003},
   url = {http://books.google.com/books?hl=en&lr=&id=KMJlC6hizIEC&oi=fnd&pg=PA323&dq=A+Family+of+Quasi-local+Mass+Functionals+with+Monotone+Flows&ots=brVkWiZvsE&sig=ovai4UODn6UJqPYkg030nn2tkFM#v=onepage&q=A},
   Key = {fds43695}
}

@article{MR2003c:53047,
   Author = {Bray, H and Finster, F},
   Title = {Curvature estimates and the Positive Mass
             Theorem},
   Journal = {Communications in Analysis and Geometry},
   Volume = {10},
   Number = {2},
   Pages = {291-306},
   Publisher = {International Press of Boston},
   Year = {2002},
   Month = {January},
   url = {http://dx.doi.org/10.4310/CAG.2002.v10.n2.a3},
   Abstract = {The Positive Mass Theorem implies that any smooth, complete,
             asymptotically flat 3-manifold with non-negative scalar
             curvature which has zero total mass is isometric to (ℝ3
             δij). In this paper, we quantify this statement using
             spinors and prove that if a complete, asymptotically flat
             manifold with non-negative scalar curvature has small mass
             and bounded isoperimetric constant, then the manifold must
             be close to (ℝ3, δij), in the sense that there is an
             upper bound for the L2 norm of the Riemannian curvature
             tensor over the manifold except for a set of small measure.
             This curvature estimate allows us to extend the case of
             equality of the Positive Mass Theorem to include non-smooth
             manifolds with generalized non-negative scalar curvature,
             which we define.},
   Doi = {10.4310/CAG.2002.v10.n2.a3},
   Key = {MR2003c:53047}
}

@article{fds287063,
   Author = {Bray, H},
   Title = {Black Holes and the Penrose Inequality in General
             Relativity},
   Journal = {Proceedings of the International Congress of
             Mathematicians},
   Volume = {2},
   Pages = {257-272},
   Booktitle = {Proceedings of the International Congress of Mathematicians,
             Beijing, China, 2002},
   Year = {2002},
   url = {http://arxiv.org/abs/math/0304261v1},
   Key = {fds287063}
}

@article{MR2002i:53073,
   Author = {Bray, H and Morgan, F},
   Title = {An isoperimetric comparison theorem for schwarzschild space
             and other manifolds},
   Journal = {Proceedings of the American Mathematical
             Society},
   Volume = {130},
   Number = {5},
   Pages = {1467-1472},
   Year = {2002},
   url = {http://www.ams.org/journals/proc/2002-130-05/S0002-9939-01-06186-X/S0002-9939-01-06186-X.pdf},
   Abstract = {We give a very general isoperimetric comparison theorem
             which, as an important special case, gives hypotheses under
             which the spherically symmetric (n - 1)-spheres of a
             spherically symmetric n-manifold are isoperimetric
             hypersurfaces, meaning that they minimize (n -
             1)-dimensional area among hypersurfaces enclosing the same
             n-volume. This result greatly generalizes the result of Bray
             (Ph.D. thesis, 1997), which proved that the spherically
             symmetric 2-spheres of 3-dimensional Schwarzschild space
             (which is defined to be a totally geodesic, space-like slice
             of the usual (3 + 1)-dimensional Schwarzsehild metric) are
             isoperimetric. We also note that this Schwarzschild result
             has applications to the Penrose inequality in general
             relativity, as described by Bray.},
   Doi = {10.1090/S0002-9939-01-06186-X},
   Key = {MR2002i:53073}
}

@article{MR2003j:83052,
   Author = {Bray, HL},
   Title = {Black Holes, Geometric Flows, and the Penrose Inequality in
             General Relativity},
   Journal = {Notices of the American Mathematical Society},
   Volume = {49},
   Number = {11},
   Pages = {1372-1381},
   Year = {2002},
   url = {http://www.ams.org/notices/200211/fea-bray.pdf},
   Key = {MR2003j:83052}
}

@article{MR2003k:83066,
   Author = {Bray, HL and Iga, K},
   Title = {Superharmonic Functions in R^n and the Penrose Inequality in
             General Relativity},
   Journal = {Communications in Analysis and Geometry},
   Volume = {10},
   Number = {5},
   Pages = {999-1016},
   Publisher = {International Press of Boston},
   Year = {2002},
   url = {http://dx.doi.org/10.4310/cag.2002.v10.n5.a5},
   Doi = {10.4310/cag.2002.v10.n5.a5},
   Key = {MR2003k:83066}
}

@article{MR2004j:53046,
   Author = {Bray, HL},
   Title = {Proof of the riemannian penrose inequality using the
             positive mass theorem},
   Journal = {Journal of Differential Geometry},
   Volume = {59},
   Number = {2},
   Pages = {177-267},
   Publisher = {International Press of Boston},
   Year = {2001},
   Month = {January},
   url = {http://www.math.duke.edu/~bray/PE/euclid.jdg.1090349428.pdf},
   Abstract = {We prove the Riemannian Penrose Conjecture, an important
             case of a conjecture [41] made by Roger Penrose in 1973, by
             defining a new flow of metrics. This flow of metrics stays
             inside the class of asymptotically flat Riemannian
             3-manifolds with nonnegative scalar curvature which contain
             minimal spheres. In particular, if we consider a Riemannian
             3-manifold as a totally geodesic submanifold of a space-time
             in the context of general relativity, then outermost minimal
             spheres with total area A correspond to apparent horizons of
             black holes contributing a mass √ A/16π, scalar curvature
             corresponds to local energy density at each point, and the
             rate at which the metric becomes flat at infinity
             corresponds to total mass (also called the ADM mass). The
             Riemannian Penrose Conjecture then states that the total
             mass of an asymptotically flat 3-manifold with nonnegative
             scalar curvature is greater than or equal to the mass
             contributed by the black holes. The flow of metrics we
             define continuously evolves the original 3-metric to a
             Schwarzschild 3-metric, which represents a spherically
             symmetric black hole in vacuum. We define the flow such that
             the area of the minimal spheres (which flow outward) and
             hence the mass contributed by the black holes in each of the
             metrics in the flow is constant, and then use the Positive
             Mass Theorem to show that the total mass of the metrics is
             nonincreasing. Then since the total mass equals the mass of
             the black hole in a Schwarzschild metric, the Riemannian
             Penrose Conjecture follows. We also refer the reader to the
             beautiful work of Huisken and Ilmanen [30], who used inverse
             mean curvature flows of surfaces to prove that the total
             mass is at least the mass contributed by the largest black
             hole. © Applied Probability Trust 2001.},
   Doi = {10.4310/jdg/1090349428},
   Key = {MR2004j:53046}
}

@incollection{MR2004j:53047,
   Author = {Bray, H and Schoen, RM},
   Title = {Recent Proofs of the Riemannian Penrose Conjecture},
   Pages = {1-36},
   Booktitle = {Current Developments in Mathematics},
   Publisher = {International Press},
   Year = {1999},
   Key = {MR2004j:53047}
}

@article{fds287082,
   Author = {Bray, H and McCormick, K and Wells, RO and Zhou, XD},
   Title = {Wavelet variations on the Shannon sampling
             theorem.},
   Journal = {Biosystems},
   Volume = {34},
   Number = {1-3},
   Pages = {249-257},
   Publisher = {Elsevier Science Ireland},
   Year = {1995},
   Month = {January},
   ISSN = {0303-2647},
   url = {http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6T2K-3YMWJCP-J&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1119554323&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=00e2987e0823dfb6839780e7c7af56ec},
   Abstract = {The Shannon sampling theorem asserts that a continuous
             square-integrable function on the real line which has a
             compactly supported Fourier transform is uniquely determined
             by its restriction to a uniform lattice of points whose
             density is determined by the support of the Fourier
             transform. This result can be extended to the wavelet
             representation of functions in two ways. First, under the
             same type of conditions as for the Shannon theorem, the
             scaling coefficients of a wavelet expansion will determine
             uniquely the given square-integrable function. Secondly, for
             a more general function, there is a unique extension from a
             given set of scaling coefficients to a full wavelet
             expansion which minimizes the local obstructions to
             translation invariance in a variational sense.},
   Doi = {10.1016/0303-2647(94)01457-i},
   Key = {fds287082}
}


%% Other   
@misc{fds287067,
   Author = {Bray, H},
   Title = {The Penrose Inequality in General Relativity and Volume
             Comparison Theorems Involving Scalar Curvature},
   Year = {1997},
   url = {arxiv:0902.3241v1},
   Key = {fds287067}
}

 

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