%% @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 Basel}, Year = {2016}, Month = {June}, ISSN = {1424-0637}, url = {http://arxiv.org/abs/1402.3287}, Abstract = {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}, Year = {2015}, Month = {December}, url = {http://arxiv.org/abs/1505.07154}, 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}, Year = {2015}, Month = {June}, ISSN = {0003-889X}, url = {http://www.springer.com/-/0/c8d239381b86496b96d95ff26f1061eb}, 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}, Year = {2015}, Month = {April}, ISSN = {0010-3616}, url = {http://arxiv.org/abs/1310.8638}, 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}, Volume = {615}, 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}, 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{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{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}, Year = {2011}, 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.}, Key = {fds287073} } @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}, Year = {2010}, ISSN = {1019-8385}, url = {http://arxiv.org/abs/1002.2814}, 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.}, Key = {fds287076} } @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 & Applied Mathematics}, Volume = {63}, Number = {9}, Pages = {1237-1247}, Year = {2010}, 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}, Volume = {27}, Number = {2}, Pages = {741-766}, Year = {2010}, 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{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}, Year = {2009}, 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}, Year = {2008}, 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}, Year = {2007}, 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}, Editor = {JC Zambrini}, Year = {2006}, Month = {January}, ISBN = {9789812704016}, 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 m c (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, m 0 (Σ) equals the Hawking mass of Σ 2 and 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 m ADM is the total mass of the complete, asymptotically flat 3-manifold with nonnegative scalar curvature, then m ADM ≥ m c (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}, 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.}, 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{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{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}, Year = {2002}, url = {http://arxiv.org/abs/math/9906047v3}, 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.}, Key = {MR2003c:53047} } @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}, Year = {2002}, 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}, Year = {2001}, 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.}, 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} } @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} } @article{fds287082, Author = {Bray, H and McCormick, K and Jr, ROW and Zhou, X-D}, Title = {Wavelet variations on the Shannon sampling theorem}, Journal = {BioSystems}, Volume = {34}, Number = {1-3}, Pages = {249-257}, Publisher = {Elsevier Science Ireland}, Year = {1995}, 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. © 1995.}, Doi = {10.1016/0303-2647(94)01457-I}, Key = {fds287082} }