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On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay. (English) Zbl 0814.53034
Let \(M^ n\) be a complete Riemannian manifold with metric \(g\), such that \(\text{Ric}_ M \geq 0\), \(\text{Vol} (B_ r(p)) \geq \Omega r^ n\) for some constant \(\Omega > 0\). Fix \(p \in M\) and \(r_ j \to \infty\). Then the sequence of pointed rescaled manifolds, \((M,p, r_ j^{-2}g)\), has a subsequence which converges in the pointed Gromov-Hausdorff topology to a length space \(M_ \infty\). A basic question concerning \(M_ \infty\) is whether or not it is unique, i.e. is \(M_ \infty\) the same up to isometry for all \(\{r_ j\}\) and all convergent subsequences. In general, it is not true even under the additional condition of quadratic curvature decay. In the present paper, for \(M\) assumed to be Ricci flat, some sufficient condition for uniqueness of \(M_ \infty\) is given. Moreover, the authors prove an analogous result for \(M\) being a Kähler manifold.

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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[1] [AA] F.J., Almgren, W.K. Allard: On the radial behaviour of minimal surfaces and the uniqueness of their tangent cones. Ann. Math.113 (1981) 215-265 · Zbl 0449.53041 · doi:10.2307/2006984
[2] [A1] M., Anderson: Ricci curvature bounds and Einstein metrics on compact manifolds, Journal A.M.S.,2 (1989) 455-490 · Zbl 0694.53045
[3] [A2] M. Anderson: Convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math.,102 (1990) 429-445 · Zbl 0711.53038 · doi:10.1007/BF01233434
[4] [AC1] M. Anderson, J. Cheeger: Finiteness theorems for manifolds with Ricci curvature andL n/2-norm of curvature bounded. J. Geom. Funct. Anal.,1 (1991) 231-252 · Zbl 0764.53026 · doi:10.1007/BF01896203
[5] [AC2] M. Anderson, J. Cheeger,C ?-compactness for manifolds with Ricci curvature and injectivity radius bounded below. J. Differ. Geom.3 (1992) 265-281 · Zbl 0774.53021
[6] [BKN] S. Bando, A. Kasue, H. Nakajima: On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Invent. Math.,97 (1989) 313-349 · Zbl 0682.53045 · doi:10.1007/BF01389045
[7] [BK] S. Bando, R. Kobayashi: Ricci flat Kähler metrics on affine algebraic manifolds, Geometry and Analysis on Manifolds, Lecture Notes in Math, Springer-Verlag (1987) 20-32
[8] [B] A. Besse: Einstein Manifolds. Ergeb. Math. Grenzgeb. Band 10, Springer, Berlin New York, 1987 · Zbl 0613.53001
[9] [C1] J. Cheeger: On the geometry of spaces with cone-like singularities. Proc. Nat. Acad. Sci.76 (1979) 2103-2106 · Zbl 0411.58003 · doi:10.1073/pnas.76.5.2103
[10] [C2] J. Cheeger: Analytic torsion and the heat equation. Ann. Math.109 (1979) 259-322 · Zbl 0412.58026 · doi:10.2307/1971113
[11] [C3] J. Cheeger: Spectral geometry of singular Riemannian spaces. J. Diff. Geom.18 (1983) 575-657 · Zbl 0529.58034
[12] [CC1] J. Cheeger, T. Colding: Almost rigidity of warped products and the structure of spaces with Ricci curvature bounded below C.R. Acad. Sci. Paris (to appear)
[13] [CC2] J. Cheeger, T. Colding: Lower bounds on Ricci curvature and the almost rigidity of warped products (preprint)
[14] [CC3] J. Cheeger, T. Colding: On the structure of spaces with Ricci curvature bounded below (to appear)
[15] [CT] J. Cheeger, G. Tian (to appear)
[16] [E] D. Ebin: The manifold of riemannian metrics. A.M.S. Proc. Sym in Pure Math. Vol XV, Global Analysis (1970) 11-40 · Zbl 0205.53702
[17] [EM] D. Ebin, J. Marsden: Groups of Diffeomorphisms and the Motion of an Incompressible Fluid. Ann. Math. (22)92 (1970) 102-163 · Zbl 0211.57401 · doi:10.2307/1970699
[18] [DNP] D.J. Duff, B.E.W. Wilson, C. Pope: Kaluzo-Klein Supergravity. Phys. Reports130 102-163 (1986) · doi:10.1016/0370-1573(86)90163-8
[19] [G] L. Gao: Convergence of Riemannian manifolds. Ricci pinching andL n/2 curvature pinching. Jour. Diff. Geom32 (1990) 349-381 · Zbl 0752.53022
[20] [GT] D. Gilbarg, N. Trudinger: Elliptic Partial Differential Equations of Second Order. Springer, New York, 1977
[21] [GH] P. Griffiths. J Harris, Principles of algebraic geometry. Wiley, New York 1978
[22] [GPL] M. Gromov, J. Lafontaine, P. Pansu: Structures Métriques Pour Les Variétés Riemanniennes. Cedic/Fernand, Nathen 1981
[23] [N] A. Nadel: Multiplier Ideal Sheaves and Existence of Kähler-Einstein Metrics of Positive Scalar Curvature. Proc. Natl. Acad. Sci. USA86 (1989)
[24] [P] G. Perelman, (unpublished)
[25] [S1] L. Simon: Asymptotics for a Class of Non-linear Evolution Equations, With Applications to Geometric Problems. Ann. of Math.118 (1983) 525-571 · Zbl 0549.35071 · doi:10.2307/2006981
[26] [S2] L. Simon: Springer Lecture Notes in Math. Springer-Verlag, 1161
[27] [T1] G. Tian: On Kähler-Einstein Metrics on Certain Kähler Manifolds WithC 1(M)>0. Invent. Math.89 (1987) 225-246 · Zbl 0599.53046 · doi:10.1007/BF01389077
[28] [T2] G. Tian: On Calabi’s Conjecture for Complex Surfaces With Positive First Chern Class. Invent. Math.101 (1990) 101-172 · Zbl 0716.32019 · doi:10.1007/BF01231499
[29] [TY] G. Tian, S.T. Yau: Complete Kähler Manifolds With Zero Ricci Curvature, II. Invent. Math.106 (1991) 27-60 · Zbl 0766.53053 · doi:10.1007/BF01243902
[30] [Y] D. Yang:L p Pinching and Compactness Theorems for Compactness Riemannian Manifolds. Séminaire de théorie Spectral et Géométric, Chambéry-Grenoble, 1987-1988) 81-89
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