%% Papers Published
@article{fds340288,
Author = {Sormani, C and Bray, HL and Minicozzi, WP and Eichmair, M and Huang,
LH and Yau, ST 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 = {11},
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 = {14571475},
Publisher = {Springer Nature},
Year = {2016},
Month = {June},
ISSN = {14240637},
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 quasilocal mass of
spacelike 2surfaces in the spacetime.},
Doi = {10.1007/s0002301504202},
Key = {fds300017}
}
@article{fds303060,
Author = {MartinezMedina, 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 = {025025},
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), UltraLight Axion
(ULA) dark matter, and BoseEinstein 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 nonspherical
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 = {10.1088/14757516/2015/12/025},
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 = {561575},
Publisher = {Springer Nature},
Year = {2015},
Month = {June},
ISSN = {0003889X},
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/s0001301507670},
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 = {285307},
Publisher = {Springer Nature},
Year = {2015},
Month = {January},
ISSN = {00103616},
url = {http://arxiv.org/abs/1310.8638},
Abstract = {© 2014, SpringerVerlag Berlin Heidelberg. We identify a
condition on spacelike 2surfaces 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 socalled “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 quasilocal mass of a spacelike
2surface contained therein.},
Doi = {10.1007/s0022001421622},
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 = {012001012001},
Publisher = {IOP Publishing},
Year = {2015},
ISSN = {17426588},
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 = {10.1088/17426596/615/1/012001},
Key = {fds287070}
}
@article{fds287066,
Author = {Bray, H and Goetz, AS},
Title = {Wave Dark Matter and the TullyFisher 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 = {525560},
Publisher = {International Press of Boston},
Year = {2013},
ISSN = {10936106},
url = {http://dx.doi.org/10.4310/AJM.2013.v17.n3.a6},
Abstract = {The Schwarzschild spacetime metric of negative mass is
wellknown 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 = {557610},
Publisher = {International Press of Boston},
Year = {2011},
Month = {January},
ISSN = {10936106},
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 SchoenYau
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 CarrascoMars
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 timeindependent 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 = {AreaMinimizing Projective Planes in 3Manifolds},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {63},
Number = {9},
Pages = {12371247},
Publisher = {WILEY},
Year = {2010},
Month = {September},
ISSN = {00103640},
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{doublestruck}P{doublestruck}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{doublestruck}P{doublestruck}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 = {741766},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2010},
Month = {June},
ISSN = {10780947},
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 spacelike
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 areaminimizing twospheres in
threemanifolds},
Journal = {Communications in Analysis and Geometry},
Volume = {18},
Number = {4},
Pages = {821830},
Publisher = {International Press of Boston},
Year = {2010},
Month = {January},
ISSN = {10198385},
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
twosphere in a threemanifold (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 = {81106},
Publisher = {Duke University Press},
Year = {2009},
Month = {May},
ISSN = {00127094},
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 = {10.1215/001270942009020},
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 = {459475},
Publisher = {Springer Nature},
Year = {2008},
Month = {June},
ISSN = {00209910},
url = {http://dx.doi.org/10.1007/s002220070102x},
Abstract = {Given a surface in an asymptotically flat 3manifold 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
3manifold. 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 SpringerVerlag.},
Doi = {10.1007/s002220070102x},
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 = {119138},
Publisher = {Springer Nature},
Year = {2007},
Month = {May},
ISSN = {00103616},
url = {http://dx.doi.org/10.1007/s0022000702039},
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 twosurfaces 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 1parameter freedom, but for the whole family, the
embedding functions satisfy a forwardbackward 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. ©
SpringerVerlag 2007.},
Doi = {10.1007/s0022000702039},
Key = {fds287084}
}
@article{fds303538,
Author = {Bray, HL},
Title = {A family of quasilocal mass functionals with monotone
flows},
Pages = {323329},
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
quasilocal mass functionals mc(Σ), 0 ≤ c ≤ ∞, which
are nondecreasing on surfaces in 3manifolds 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 3manifold 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 3Manifolds with Yamabe Invariant
Greater than RP^3},
Journal = {Annals of Mathematics},
Volume = {159},
Number = {1},
Pages = {407424},
Publisher = {Annals of Mathematics, Princeton U},
Year = {2004},
Month = {January},
url = {http://annals.math.princeton.edu/2004/1591/p09},
Abstract = {In this paper we compute the σinvariants (sometimes also
called the smooth Yamabe invariants) of RP3 and RP2×S1
(which are equal) and show that the only prime 3manifolds
with larger σinvariants are S3, S2×S1, and S2×~S1 (the
nonorientable S2 bundle over S1). More generally, we show
that any 3manifold with σinvariant 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 3manifolds with σinvariant
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:grqc/0312047v2},
Key = {fds287060}
}
@article{fds43695,
Author = {H.L. Bray},
Title = {A Family of Quasilocal Mass Functionals with Monotone
Flows},
Booktitle = {Proceedings of the 14th International Congress on
Mathematical Physics, Lisbon, Portugal, 2003},
Editor = {JeanClaude Zambrini},
Year = {2003},
url = {http://books.google.com/books?hl=en&lr=&id=KMJlC6hizIEC&oi=fnd&pg=PA323&dq=A+Family+of+Quasilocal+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 = {291306},
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 3manifold with nonnegative 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 nonnegative 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 nonsmooth
manifolds with generalized nonnegative 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 = {257272},
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 = {14671472},
Year = {2002},
url = {http://www.ams.org/journals/proc/200213005/S000299390106186X/S000299390106186X.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 nmanifold are isoperimetric
hypersurfaces, meaning that they minimize (n 
1)dimensional area among hypersurfaces enclosing the same
nvolume. This result greatly generalizes the result of Bray
(Ph.D. thesis, 1997), which proved that the spherically
symmetric 2spheres of 3dimensional Schwarzschild space
(which is defined to be a totally geodesic, spacelike 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/S000299390106186X},
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 = {13721381},
Year = {2002},
url = {http://www.ams.org/notices/200211/feabray.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 = {9991016},
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 = {177267},
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
3manifolds with nonnegative scalar curvature which contain
minimal spheres. In particular, if we consider a Riemannian
3manifold as a totally geodesic submanifold of a spacetime
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 3manifold 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 3metric to a
Schwarzschild 3metric, 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 = {136},
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 = {13},
Pages = {249257},
Publisher = {Elsevier Science Ireland},
Year = {1995},
Month = {January},
ISSN = {03032647},
url = {http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6T2K3YMWJCPJ&_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
squareintegrable 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 squareintegrable 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 = {10.1016/03032647(94)01457i},
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}
}
