On boundary value problems for Einstein metrics

Anderson, Michael T¹

¹ Dept of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794-3651, USA

Geometry & topology, Tome 12 (2008) no. 4, pp. 2009-2045.

Résumé

On any given compact manifold $M^{n+1}$ with boundary $\partial M$, it is proved that the moduli space $\mathcal{ℰ}$ of Einstein metrics on $M$, if non-empty, is a smooth, infinite dimensional Banach manifold, at least when ${\pi }_1\left(M,\partial M\right)=0$. Thus, the Einstein moduli space is unobstructed. The usual Dirichlet and Neumann boundary maps to data on $\partial M$ are smooth, but not Fredholm. Instead, one has natural mixed boundary-value problems which give Fredholm boundary maps.

These results also hold for manifolds with compact boundary which have a finite number of locally asymptotically flat ends, as well as for the Einstein equations coupled to many other fields.

Détail

DOI : 10.2140/gt.2008.12.2009

Keywords: Einstein metrics, elliptic boundary, value problems

Bibliographie

[1] S Agmon, A Douglis, L Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I and II, Comm. Pure Appl. Math. 12 and 17 (1964)

[2] M Anderson, Geometric aspects of the AdS/CFT correspondence, from: "AdS/CFT correspondence: Einstein metrics and their conformal boundaries", IRMA Lect. Math. Theor. Phys. 8, Eur. Math. Soc., Zürich (2005) 1

[3] M T Anderson, M Herzlich, Unique continuation results for Ricci curvature and applications, J. Geom. Phys. 58 (2008) 179

[4] M Anderson, A Katsuda, Y Kurylev, M Lassas, M Taylor, Boundary regularity for the Ricci equation, geometric convergence, and Gel'fand's inverse boundary problem, Invent. Math. 158 (2004) 261

[5] A L Besse, Einstein manifolds, Ergebnisse series 10, Springer (1987)

[6] D G Ebin, The manifold of Riemannian metrics, from: "Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968)", Amer. Math. Soc. (1970) 11

[7] J Eells, L Lemaire, Deformations of metrics and associated harmonic maps, from: "Geometry and analysis", Indian Acad. Sci. (1980) 33

[8] D Gilbarg, N S Trudinger, Elliptic partial differential equations of second order, Grundlehren series 224, Springer (1983)

[9] R S Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. $($N.S.$)$ 7 (1982) 65

[10] S W Hawking, G F R Ellis, The large scale structure of space-time, Cambridge University Press (1973)

[11] S Kobayashi, K Nomizu, Foundations of Differential Geometry, Volume 1, John Wiley, New York (1963)

[12] N Koiso, Rigidity and infinitesimal deformability of Einstein metrics, Osaka J. Math. 19 (1982) 643

[13] C Lebrun, M Wang, Surveys in differential geometry: essays on Einstein manifolds, Surveys in Differential Geometry, VI, International Press (1999)

[14] A Marini, Dirichlet and Neumann boundary value problems for Yang–Mills connections, Comm. Pure Appl. Math. 45 (1992) 1015

[15] R Mazzeo, R B Melrose, Pseudodifferential operators on manifolds with fibred boundaries, Asian J. Math. 2 (1998) 833

[16] C B Morrey Jr., Multiple integrals in the calculus of variations, Grundlehren series 130, Springer, New York (1966)

[17] J Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. $(2)$ 63 (1956) 20

[18] L Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953) 337

[19] J M Schlenker, Einstein manifolds with convex boundaries, Comment. Math. Helv. 76 (2001) 1