On Expansions of Ricci Flat ALE Metrics in Harmonic Coordinates About the Infinity

Abstract

In this paper, we study the expansions of Ricci flat metrics in harmonic coordinates about the infinity of ALE (Asymptotically Local Euclidean) manifolds.

Keywords

ALE / Ricci flat / Harmonic coordinates / Asymptotic expansion

References

[1] Anderson MT. Ricci curvature bounds and Einstein metrics on compact manifolds. J. Am. Math. Soc.. 1989, 2 3 455-490

[2] Axler S, Bourdon P, Ramey W. Harmonic Function Theory, Graduate Texts in Mathematics. 2001 2 New York: Springer

[3] Bando S. Bubbling out of einstein manifolds. Tohoku Math. J. Second Ser.. 1990, 42 2 205-216

[4] 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.. 1989, 97 2 313-349

[5] Bartnik R. The mass of an asymptotically flat manifold. Commun. Pure Appl. Math.. 1986, 39 5 661-693

[6] Chruściel PT, Delay E, Lee JM, Skinner DN . Boundary regularity of conformally compact Einstein metrics. J. Differ. Geom.. 2005, 69 1 111-136

[7] Fefferman C, Graham CR. The Ambient Metric (AM-178). 2011 Princeton: Princeton University Press

[8] Fefferman Charles L. Monge-Ampere Equations, the Bergman Kernel, and Geometry of Pseudoconvex Domains. The Annals of Mathematics. 1976, 103 3 395

[9] Graham, C.R.: Volume and area renormalizations for conformally compact Einstein metrics. arXiv preprint arXiv:math/9909042 (1999)

[10] Han, Q., Jiang, X.: Boundary expansions for minimal graphs in the hyperbolic space. arXiv preprint arXiv:1412.7608 (2014)

[11] Han, Q. Wang, Y.: Boundary expansions for constant mean curvature surfaces in the hyperbolic space. arXiv preprint arXiv:1608.07803 (2016)

[12] Helliwell DW. Boundary regularity for conformally compact Einstein metrics in even dimensions. Commun. Partial Differ. Equ.. 2008, 33 5 842-880

[13] Jeffres T, Mazzeo R, Rubinstein YA. Kähler–Einstein metrics with edge singularities. Ann. Math.. 2016, 183 1 95-176

[14] Jian H, Wang X. Optimal boundary regularity for nonlinear singular elliptic equations. Adv. Math.. 2014, 251 111-126

[15] Jian H, Wang X-J . Bernstein theorem and regularity for a class of Monge–Ampère equations. J. Differ. Geom.. 2013, 93 3 431-469

[16] Kichenassamy S. On a conjecture of Fefferman and Graham. Adv. Math.. 2004, 184 2 268-288

[17] Korevaar N, Mazzeo R, Pacard F, Schoen R. Refined asymptotics for constant scalar curvature metrics with isolated singularities. Invent. Math.. 1999, 135 2 233-272

[18] Lee J, Melrose R. Boundary behavior of the complex Monge–Ampere equation. Acta Math.. 1982, 148 1 159-192

[19] Lin FH. Asymptotic behavior of area-minimizing currents in hyperbolic space. Commun. Pure Appl. Math.. 1989, 42 3 229-242

[20] Lin FH. On the Dirichlet problem for minimal graphs in hyperbolic space. Invent. Math.. 1989, 96 3 593-612

[21] Mazzeo R. Regularity for the singular Yamabe problem. Indiana Univ. Math. J.. 1991, 40 4 1277-1299

[22] Meyers N. An expansion about infinity for solutions of linear elliptic equations. J. Math. Mech.. 1963, 12 247-264

[23] Nakajima H. Hausdorff convergence of Einstein 4-manifolds. J. Fac. Sci. Univ. Tokyo. 1988, 35 411-424

[24] Ni L, Shi Y, Tam L-F. Ricci flatness of asymptotically locally Euclidean metrics. Trans. Am. Math. Soc.. 2003, 355 5 1933-1959

[25] Yin H. Boundary regularity of harmonic maps from hyperbolic space into nonpositively curved manifolds. Pac. J. Math. 2007, 232 491-509

[26] Yin, H.: Analysis aspects of Ricci flow on conical surfaces. arXiv preprint arXiv:1605.08836 (2016)

[27] Yin, H., Zheng, K.: Expansion formula for complex Monge–Ampere equation along cone singularities. arXiv preprint arXiv:1609.03111 (2016)