# Hubert Bray

## Professor of Mathematics

### Overview

Professor Bray uses differential geometry to understand general relativity, and general relativity to motivate interesting problems in differential geometry. In 2001, he published his proof of the Riemannian Penrose Conjecture about the mass of black holes using geometric ideas related to minimal surfaces, scalar curvature, conformal geometry, geometric flows, and harmonic functions. He is also interested in the large-scale unexplained curvature of the universe, otherwise known as dark matter, which makes up most of the mass of galaxies. Professor Bray has proposed geometric explanations for dark matter which he calls "wave dark matter," which motivate very interesting questions about geometric partial differential equations.

Professor Bray has supervised 8 math Ph.D. graduates at Duke from 2006 to 2017. He is currently supervising one math Ph.D. student and one physics Ph.D. student. His most recent Ph.D. graduate, Henri Roesch, proved a Null Penrose Conjecture, open since 1973, as his thesis. While the physical motivation about the mass of black holes is the same as for the Riemannian Penrose Conjecture, the geometry involved is almost unrecognizably different, and may be viewed as a fundamental result about null geometry.

Bray, H. *On Wave Dark Matter, Shells in Elliptical Galaxies, and the Axioms of General Relativity*. Dec. 2012.

Bray, H. L., and M. A. Khuri. “P. D. E. 'S which imply the penrose conjecture.” *Asian Journal of Mathematics*, vol. 15, no. 4, Jan. 2011, pp. 557–610. *Scopus*, doi:10.4310/AJM.2011.v15.n4.a5.
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Bray, H., et al. “Area-Minimizing Projective Planes in 3-Manifolds.” *Communications on Pure and Applied Mathematics*, vol. 63, no. 9, Sept. 2010, pp. 1237–47. *Scopus*, doi:10.1002/cpa.20319.
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Bray, H. L., and M. A. Khuri. “A jang equation approach to the penrose inequality.” *Discrete and Continuous Dynamical Systems*, vol. 27, no. 2, June 2010, pp. 741–66. *Scopus*, doi:10.3934/dcds.2010.27.741.
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Bray, H., et al. “Rigidity of area-minimizing two-spheres in three-manifolds.” *Communications in Analysis and Geometry*, vol. 18, no. 4, Jan. 2010, pp. 821–30. *Scopus*, doi:10.4310/CAG.2010.v18.n4.a6.
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Bray, H. L., and D. A. Lee. “On the Riemannian Penrose inequality in dimensions less than eight.” *Duke Mathematical Journal*, vol. 148, no. 1, May 2009, pp. 81–106. *Scopus*, doi:10.1215/00127094-2009-020.
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Bray, H., and P. Miao. “On the capacity of surfaces in manifolds with nonnegative scalar curvature.” *Inventiones Mathematicae*, vol. 172, no. 3, June 2008, pp. 459–75. *Scopus*, doi:10.1007/s00222-007-0102-x.
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Bray, H., et al. “Generalized inverse mean curvature flows in spacetime.” *Communications in Mathematical Physics*, vol. 272, no. 1, May 2007, pp. 119–38. *Scopus*, doi:10.1007/s00220-007-0203-9.
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Bray, H. L., and A. Neves. “Classification of Prime 3-Manifolds with Yamabe Invariant Greater than RP^3.” *Annals of Mathematics*, vol. 159, no. 1, Jan. 2004, pp. 407–24.

Bray, H. L., and K. Iga. “Superharmonic Functions in R^n and the Penrose Inequality in General Relativity.” *Communications in Analysis and Geometry*, vol. 10, no. 5, 2002, pp. 999–1016.