Local existence and uniqueness for exterior static vacuum Einstein metrics

Anderson, Michael

doi:10.1090/S0002-9939-2015-12486-0

Local existence and uniqueness for exterior static vacuum Einstein metrics
HTML articles powered by AMS MathViewer

by Michael T. Anderson

Proc. Amer. Math. Soc. 143 (2015), 3091-3096

DOI: https://doi.org/10.1090/S0002-9939-2015-12486-0

Published electronically: February 5, 2015

PDF | Request permission

Abstract:

We study solutions to the static vacuum Einstein equations on domains of the form $M \simeq \mathbb {R}^{3}\setminus B$ with prescribed Bartnik data $(\gamma , H)$ on the inner boundary $\partial M$. It is proved that for any smooth boundary data $(\gamma , H)$ close to standard round data on the unit sphere $(\gamma _{+1}, 2)$, there exists a unique asymptotically flat solution of the static vacuum Einstein equations realizing the boundary data $(\gamma , H)$ which is close to the standard flat solution.

References

Michael T. Anderson, On boundary value problems for Einstein metrics, Geom. Topol. 12 (2008), no. 4, 2009–2045. MR 2431014, DOI 10.2140/gt.2008.12.2009
Michael T. Anderson, Extension of symmetries on Einstein manifolds with boundary, Selecta Math. (N.S.) 16 (2010), no. 3, 343–375. MR 2734335, DOI 10.1007/s00029-010-0028-9
Michael T. Anderson and Marcus A. Khuri, On the Bartnik extension problem for the static vacuum Einstein equations, Classical Quantum Gravity 30 (2013), no. 12, 125005, 33. MR 3064190, DOI 10.1088/0264-9381/30/12/125005
Robert Bartnik, New definition of quasilocal mass, Phys. Rev. Lett. 62 (1989), no. 20, 2346–2348. MR 996396, DOI 10.1103/PhysRevLett.62.2346
Robert Bartnik, Mass and 3-metrics of non-negative scalar curvature, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 231–240. MR 1957036
R. Beig and W. Simon, Proof of a multipole conjecture due to Geroch, Comm. Math. Phys. 78 (1980/81), no. 1, 75–82. MR 597031
Piotr T. Chruściel, Gregory J. Galloway, and Daniel Pollack, Mathematical general relativity: a sampler, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 4, 567–638. MR 2721040, DOI 10.1090/S0273-0979-2010-01304-5
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Pengzi Miao, On existence of static metric extensions in general relativity, Comm. Math. Phys. 241 (2003), no. 1, 27–46. MR 2013750, DOI 10.1007/s00220-003-0925-2
P. Miao, personal communication.