×

zbMATH — the first resource for mathematics

On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. (English) Zbl 0682.53045
The paper is concerned with the construction of certain Euclidean-like coordinate systems at infinity of complete Riemannian manifolds with curvature decay (for the Riemann and Ricci curvature tensors) and volume ascent of balls of prescribed order \(<-2\) and n, respectively. In particular, a conjecture of H. Nakajima on Ricci-flat manifolds of dimension \(\geq 4\) can be answered with these methods [J. Fac. Sci., Univ. Tokyo, Sect. I A 35, 411-424 (1988; Zbl 0655.53037)]. The proofs are highly technical, consisting of numerous estimates, and relying on results on Laplacian and Hessian comparison theorems, harmonic coordinates, Gromov’s convergence theorem and J. Moser’s iteration scheme.
Reviewer: R.Walter

MSC:
53C20 Global Riemannian geometry, including pinching
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [An] Anderson, M.: Ricci curvature bounds and Einstein metrics on compact manifolds (Preprint) · Zbl 0694.53045
[2] [AJ] Atiyah, M.F., Jones, J.D.S.: Topological aspects of Yang-Mills theory. Comm. Math. Phys.61, 97-118 (1978) · Zbl 0387.55009
[3] [BK] Bando, S., Kobayashi, R.: Ricci-flat Kähler metrics on affine algebraic manifolds, II (Preprint) · Zbl 0701.53083
[4] [Ba] Bartnik, R.: The mass of an asymptotically flat manifold. Comm. Pure Appl. Math.34, 661-693 (1986) · Zbl 0598.53045
[5] [BC] Bishop, R., Crittenden, R.: Geometry of Manifolds. Academic Press: New York 1964 · Zbl 0132.16003
[6] [BL] Bourguignon, J.P., Lawson, H.B.: Stability and isolation phenomena for Yang-Mills fields. Comm. Math. Phys.79, 189-230 (1981) · Zbl 0475.53060
[7] [Ca] Calabi, E.: Métriques kählériennes et fibrés holomorphes. Ann. Sci. Ec. Norm. Super.12, 269-294 (1979)
[8] [CG] Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differ. Geom.6, 119-128 (1971) · Zbl 0223.53033
[9] [CGT] Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom.17, 15-53 (1982) · Zbl 0493.53035
[10] [CY] Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure. Appl. Math.28, 333-354 (1975) · Zbl 0312.53031
[11] [Cr] Croke, C.: Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. Ec. Norm. Super.13, 419-435 (1980) · Zbl 0465.53032
[12] [DK] De Turck, D., Kazdan, J.: Some regularity problems in Riemannian geometry. Ann. Sci. Ec. Norm. Sup.14, 249-260 (1981)
[13] [GT] Gilbarg, D., Trudinger, N.S.: Partial differential equations of second order, second edition. Berlin Heidelberg New York: Springer 1983 · Zbl 0562.35001
[14] [GW] Greene, R.E., Wu, H.: Lipschitz convergence of Riemannian manifolds. Pac. J. Math.131, 119-141 (1988) · Zbl 0646.53038
[15] [GKM] Gromoll, D., Klingenberg, W., Meyer, W.: Riemannsche Geometrie im Großen. Berlin Heidelberg New York: Springer 1968 · Zbl 0155.30701
[16] [Gr] Gromov, M.: Structures métrique pour les varitétés riemanniennes. Redigé par J. Lafontaine et P. Pansu, Textes Math. No. 1, Cedic/Fernand Nathan: Paris 1981
[17] [Jo] Jost, J.: Harmonic mappings between Riemannian manifolds. Proc. Centre Math. Anal., Australien Nat. University 1983 · Zbl 0542.58001
[18] [K1] Kasue, A.: A Laplacian comparison theorem and function theoretic properties of a complete Riemannian manifold. Jpn. J. Math.8, 309-341 (1982) · Zbl 0518.53048
[19] [K2] Kasue, A.: A compactification of a manifold with asymptotically nonnegative curvature. Ann. Sci. Ec. Norm. Sup. (to appear)
[20] [K3] Kasue, A.: A convergence theorem for Riemannian manifolds and some applications. Nagoya Math. J. (to appear)
[21] [Kr] Kronheimer, P.B.: ALE gravitational instantons. Thesis, Oxford University 1986
[22] [LP] Lee, J., Parker, T.: The Yamabe problem. Bull. Am. Math. Soc. N.S.17, 37-91 (1987) · Zbl 0633.53062
[23] [Mo] Mok, N.: An embedding theorem of complete Kähler manifolds of positive bisectional curvature onto an affine algebraic varieties. Bull. Soc. Math. Fr.112, 197-258 (1984) · Zbl 0536.53062
[24] [MSY] Mok, N., Siu, Y.T., Yau, S.T.: The Poincare-Lelong equation on complete Kähler manifolds. Comp. Math.44, 183-218 (1981) · Zbl 0531.32007
[25] [Ng] Nakagawa, H.: Taiiki no Riemann Kikagaku (Riemannian geometry in the large). Kaigai Shuppan Boeki: Tokyo 1977 (in Japanese)
[26] [N1] Nakajima, H.: Removable singularities for Yang-Mills connections in higher dimensions. J. Fac. Sci. Univ. Tokyo34, 299-307 (1987) · Zbl 0637.58026
[27] [N2] Nakajima, H.: Hausdorff convergence of Einstein 4-manifolds. J. Fac. Sci. Univ. Tokyo35, 411-424 (1988) · Zbl 0655.53037
[28] [OS] Otway, T.H., Sibner, L.M.: Point singularities of coupled gauge fields with low energy. Comm. Math. Phys.111, 275-279 (1987) · Zbl 0628.53078
[29] [Pe] Peters, S.: Convergence of Riemannian manifolds. Compos. Math.62, 3-16 (1987) · Zbl 0618.53036
[30] [Pr] Price, P.: A monotonicity formula for Yang-Mills fields. Manuscr. Math.43, 131-166 (1984) · Zbl 0521.58024
[31] [Sc] Schoen, R.: Conformal deformation of Riemannian metrics and constant scalar curvature. J. Differ. Geom.20, 479-495 (1984) · Zbl 0576.53028
[32] [SSY] Schoen, R., Simon, L., Lau, S.T.: Curvature estimates for minimal hypersurfaces. Acta Math.334, 275-288 (1975) · Zbl 0323.53039
[33] [Si] Sibner, L.: The isolated point singularity problem for the coupled Yang-Mills equations in higher dimensions. Math. Ann.271, 125-131 (1985) · Zbl 0558.35073
[34] [Sm] Simon, L.: Lectures on geometric measure theory. Proc. Centre Math. Anal., Australian Nat. University 1983 · Zbl 0546.49019
[35] [Su] Siu, Y.T.: The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group. Ann. Math.127, 585-627 (1988) · Zbl 0651.53035
[36] [Ti] Tian, G.: On Kähler-Einstein metrics on certain Kähler manifolds withc 1 (M)>0. Invent. Math.89, 225-246 (1987) · Zbl 0599.53046
[37] [Uh] Uhlenbeck, K.: Removable singularities in Yang-Mills fields. Comm. Math. Phys.83, 11-30 (1982) · Zbl 0491.58032
[38] [W] Warner, F.W.: Extension of the Rauch comparison theorem to submanifolds. Trans. Am. Math. Soc.122, 341-356 (1966) · Zbl 0139.15601
[39] [Wu] Wu, H.: An elementary method in the study of nonnegative curvature. Acta. Math.142, 57-78 (1979) · Zbl 0403.53022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.