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A note on asymptotically flat metrics on $\mathbb{R}^3$ which are scalar-flat and admit minimal spheres


Author: Justin Corvino
Journal: Proc. Amer. Math. Soc. 133 (2005), 3669-3678
MSC (2000): Primary 53C21, 83C99
DOI: https://doi.org/10.1090/S0002-9939-05-07926-8
Published electronically: June 8, 2005
MathSciNet review: 2163606
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Abstract: We use constructions by Miao and Chrusciel-Delay to produce asymptotically flat metrics on $\mathbb{R}^3$ which have zero scalar curvature and multiple stable minimal spheres. Such metrics are solutions of the time-symmetric vacuum constraint equations of general relativity, and in this context the horizons of black holes are stable minimal spheres. We also note that under pointwise sectional curvature bounds, asymptotically flat metrics of nonnegative scalar curvature and small mass do not admit minimal spheres, and hence are topologically $\mathbb{R}^3$.


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Additional Information

Justin Corvino
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
Address at time of publication: Department of Mathematics, Lafayette College, Easton, Pennsylvania 18042
Email: corvinoj@lafayette.edu

DOI: https://doi.org/10.1090/S0002-9939-05-07926-8
Received by editor(s): May 24, 2004
Received by editor(s) in revised form: August 13, 2004
Published electronically: June 8, 2005
Additional Notes: The author was partly supported by an NSF postdoctoral research fellowship
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2005 American Mathematical Society

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