Abstract
We study the existence and uniqueness of solutions to the static vacuum Einstein equations in bounded domains, satisfying the Bartnik boundary conditions of prescribed metric and mean curvature on the boundary.
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Akbar M., Gibbons G.: Ricci-flat metrics with U(1) action and the Dirichlet boundary-value problem in Riemannian quantum gravity and isoperimetric inequalities. Class. Quantum Gravity 20, 1787–1822 (2003)
Ambrosetti, A., Prodi, G.: A Primer of Nonlinear Analysis. Cambridge Studies in Advanced Mathematics, vol. 34, Cambridge University Press, Cambridge(1993)
Anderson M.: Ricci curvature bounds and Einstein metrics on compact manifolds. J. Am. Math. Soc. 2, 455–490 (1989)
Anderson M.: On stationary solutions to the vacuum Einstein equations. Ann. Henri Poincaré 1, 977–994 (2000)
Anderson M.: On the structure of solutions to the static vacuum Einstein equations. Ann. Henri Poincaré 1, 995–1042 (2000)
Anderson M.: On boundary value problems for Einstein metrics. Geom. Topol. 12, 2009–2045 (2008)
Anderson, M.: Boundary value problems for metrics on 3-manifolds. In: Dai, X., Rong, X. (eds.) Metric and Differential Geometry in Honor of J. Cheeger, pp. 3–17. Birkhäuser Verlag, Basel (2012)
Anderson, M.: Conformal immersions with prescribed mean curvature in \({\mathbb{R}^{3}}\). Preprint, (2012). arXiv:1204.5225 [math.DG]
Anderson, M., Herzlich, M.: Unique continuation results for Ricci curvature and applications. J. Geom. Phys. 58, 179–207 (2008) (Erratum, ibid., 60, 1062–1067 (2010))
Anderson M., Katsuda A., Kurylev Y., Lassas M., Taylor M.: Boundary regularity for the Ricci equation, geometric convergence and Gel’fand’s inverse boundary problem. Invent. Math. 158, 261–321 (2004)
Anderson M., Khuri M.: On the Bartnik extension problem for the static vacuum Einstein equations. Class. Quantum Gravity 30, 125005 (2013)
Bando S., Kasue A., Nakajima H.: On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Invent. Math. 97, 313–349 (1989)
Bartnik R.: New definition of quasi-local mass. Phys. Rev. Lett. 62, 2346–2348 (1989)
Bartnik, R.: Mass and 3-metrics of non-negative scalar curvature. In: Proceedings of the International Congress Mathematics, vol II, pp. 231–240. Higher Ed. Press, Beijing (2002)
Besse A.: Einstein Manifolds. Springer, Berlin (1987)
Böhme R., Tromba A.: The index theorem for classical minimal surfaces. Ann. Math. 113, 447–499 (1981)
Bott R.: Lectures on Morse theory, old and new. Bull. Am. Math. Soc. 7, 331–358 (1982)
Galloway G.: On the topology of black holes. Commun. Math. Phys. 151, 53–66 (1993)
Gilbarg D., Trudinger N.: Elliptic Partial Differential Equations of Second Order, 2nd Edition. Springer, New York (1983)
Jauregui J.: Fill-ins of nonnegative scalar curvature, static metrics and quasi-local mass. Pacif. J. Math. 261, 417–444 (2013)
Jauregui J., Miao P., Tam L.-F.: Extensions and fill-ins with nonnegative scalar curvature. Class. Quantum Gravity 30, 195007 (2013)
Knox, K.: A compactness theorem for Riemannian manifolds with boundary and applications. Preprint (2012). arXiv:1211.6210 [math.DG]
Morrey C.B. Jr.: Multiple Integrals in the Calculus of Variations. Springer, New York (1966)
Nirenberg L.: Variational and topological methods in nonlinear problems. Bull. Am. Math. Soc. 4, 267–302 (1981)
Petersen P.: Riemannian Geometry, 2nd edn. Springer, New York (2006)
Rosenberg, J., Stolz, S.: Metrics of positive scalar curvature and connections with surgery. In: Surveys on Surgery Theory, Annals of Mathematical Studies, vol. 149, pp. 353–386. Princeton University Press (2001)
Sabitov I.K.H.: Isometric surfaces with a common mean curvature and the problem of Bonnet pairs. Sbornik: Mathematics 203, 111–152 (2012)
Shi Y.-G., Tam L.-F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differ. Geom. 62, 79–125 (2002)
Smale S.: An infinite dimensional version of Sard’s theorem. Am. J. Math. 87, 861–866 (1965)
Taylor, M.: Partial Differential Equations II. Applied Mathematical Sciences, vol. 116. Springer, New York (1996)
Tod P.: Spatial metrics which are static in many ways. Gen. Rel. Grav. 32, 2079–2090 (2000)
Tromba A.: Degree theory on oriented infinite dimensional varieties and the Morse number of minimal surfaces spanning a curve in \({{\mathbb{R}}^{n}}\). Trans. Am. Math. Soc. 200, 385–415 (1985)
White B.: The space of m-dimensional surfaces that are stationary for a parametric elliptic functional. Ind. Univ. Math. J. 36, 567–602 (1987)
White B.: The space of minimal submanifolds for varying Riemannian metrics. Ind. Univ. Math. J. 40, 161–200 (1991)
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Communicated by James A. Isenberg.
Partially supported by NSF grant DMS 1205947.
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Anderson, M.T. Static Vacuum Einstein Metrics on Bounded Domains. Ann. Henri Poincaré 16, 2265–2302 (2015). https://doi.org/10.1007/s00023-014-0367-8
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DOI: https://doi.org/10.1007/s00023-014-0367-8