Ricci flatness of asymptotically locally Euclidean metrics
HTML articles powered by AMS MathViewer
- by Lei Ni, Yuguang Shi and Luen-Fai Tam PDF
- Trans. Amer. Math. Soc. 355 (2003), 1933-1959 Request permission
Abstract:
In this article we study the metric property and the function theory of asymptotically locally Euclidean (ALE) Kähler manifolds. In particular, we prove the Ricci flatness under the assumption that the Ricci curvature of such manifolds is either nonnegative or nonpositive. The result provides a generalization of previous gap type theorems established by Greene and Wu, Mok, Siu and Yau, etc. It can also be thought of as a general positive mass type result. The method also proves the Liouville properties of plurisubharmonic functions on such manifolds. We also give a characterization of Ricci flatness of an ALE Kähler manifold with nonnegative Ricci curvature in terms of the structure of its cone at infinity.References
- Uwe Abresch, Lower curvature bounds, Toponogov’s theorem, and bounded topology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 4, 651–670. MR 839689, DOI 10.24033/asens.1499
- M.-T. Anderson, The compactification of a minimal submanifold in Euclidean spaces by the Gauss map, IHES preprint, 1984.
- Shigetoshi Bando, Atsushi Kasue, and Hiraku Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), no. 2, 313–349. MR 1001844, DOI 10.1007/BF01389045
- Robert Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), no. 5, 661–693. MR 849427, DOI 10.1002/cpa.3160390505
- Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
- Eric Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math. 37 (1976), no. 1, 1–44. MR 445006, DOI 10.1007/BF01418826
- E. Calabi, Métriques kählériennes et fibrés holomorphes, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 2, 269–294 (French). MR 543218, DOI 10.24033/asens.1367
- Jeff Cheeger and Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119–128. MR 303460
- B.-L. Chen and X.-P. Zhu, On complete Kähler manifolds with positive bisectional curvature, preprint.
- S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354. MR 385749, DOI 10.1002/cpa.3160280303
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- Günter Drees, Asymptotically flat manifolds of nonnegative curvature, Differential Geom. Appl. 4 (1994), no. 1, 77–90. MR 1264910, DOI 10.1016/0926-2245(94)00002-6
- Tohru Eguchi and Andrew J. Hanson, Self-dual solutions to Euclidean gravity, Ann. Physics 120 (1979), no. 1, 82–106. MR 540896, DOI 10.1016/0003-4916(79)90282-3
- G.-W Gibbons and S. Hawking, Gravitational multi-instantons, Phys. Lett. B78 (1978), 430-432.
- Hans Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368 (German). MR 137127, DOI 10.1007/BF01441136
- R. E. Greene, A genealogy of noncompact manifolds of nonnegative curvature: history and logic, Comparison geometry (1997), 99–134.
- Robert E. Greene, Peter Petersen, and Shun-Hui Zhu, Riemannian manifolds of faster-than-quadratic curvature decay, Internat. Math. Res. Notices 9 (1994), 363ff., approx. 16 pp.}, issn=1073-7928, review= MR 1301436, doi=10.1155/S1073792894000401, DOI 10.1155/S1073792894000401
- R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. MR 521983, DOI 10.1007/BFb0063413
- R. E. Greene and H. Wu, $C^{\infty }$ approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 1, 47–84. MR 532376, DOI 10.24033/asens.1361
- R. E. Greene and H. Wu, Gap theorems for noncompact Riemannian manifolds, Duke Math. J. 49 (1982), no. 3, 731–756. MR 672504, DOI 10.1215/S0012-7094-82-04937-7
- Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
- N. J. Hitchin, Polygons and gravitons, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 3, 465–476. MR 520463, DOI 10.1017/S0305004100055924
- Dominic Joyce, Asymptotically locally Euclidean metrics with holonomy $\textrm {SU}(m)$, Ann. Global Anal. Geom. 19 (2001), no. 1, 55–73. MR 1824171, DOI 10.1023/A:1006622430781
- Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
- Atsushi Kasue and Kunio Sugahara, Gap theorems for certain submanifolds of Euclidean spaces and hyperbolic space forms, Osaka J. Math. 24 (1987), no. 4, 679–704. MR 927056
- P. B. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), no. 3, 665–683. MR 992334, DOI 10.4310/jdg/1214443066
- S. Minakshi Sundaram, On non-linear partial differential equations of the parabolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 479–494. MR 0000088, DOI 10.1007/BF03046993
- Peter Li and Richard Schoen, $L^p$ and mean value properties of subharmonic functions on Riemannian manifolds, Acta Math. 153 (1984), no. 3-4, 279–301. MR 766266, DOI 10.1007/BF02392380
- Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI 10.1007/BF02399203
- W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43–77. MR 161019
- Ngaiming Mok, An embedding theorem of complete Kähler manifolds of positive bisectional curvature onto affine algebraic varieties, Bull. Soc. Math. France 112 (1984), no. 2, 197–250 (English, with French summary). MR 788968
- Helen Moore, Minimal submanifolds with finite total scalar curvature, Indiana Univ. Math. J. 45 (1996), no. 4, 1021–1043. MR 1444477, DOI 10.1512/iumj.1996.45.1127
- Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511, DOI 10.1007/978-3-540-69952-1
- Jürgen Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. MR 159138, DOI 10.1002/cpa.3160140329
- J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of $R^{n}$, Comm. Pure Appl. Math. 26 (1973), 361–379. MR 344978, DOI 10.1002/cpa.3160260305
- Ngaiming Mok, Yum Tong Siu, and Shing Tung Yau, The Poincaré-Lelong equation on complete Kähler manifolds, Compositio Math. 44 (1981), no. 1-3, 183–218. MR 662462
- James Morrow and Kunihiko Kodaira, Complex manifolds, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1971. MR 0302937
- Raghavan Narasimhan, The Levi problem for complex spaces, Math. Ann. 142 (1960/61), 355–365. MR 148943, DOI 10.1007/BF01451029
- Lei Ni, Vanishing theorems on complete Kähler manifolds and their applications, J. Differential Geom. 50 (1998), no. 1, 89–122. MR 1678481
- Lei Ni, Yuguang Shi, and Luen-Fai Tam, Poisson equation, Poincaré-Lelong equation and curvature decay on complete Kähler manifolds, J. Differential Geom. 57 (2001), no. 2, 339–388. MR 1879230
- Wan-Xiong Shi, Ricci flow and the uniformization on complete noncompact Kähler manifolds, J. Differential Geom. 45 (1997), no. 1, 94–220. MR 1443333
- Richard Schoen and Shing Tung Yau, Proof of the positive mass theorem. II, Comm. Math. Phys. 79 (1981), no. 2, 231–260. MR 612249, DOI 10.1007/BF01942062
- Yum Tong Siu and Shing Tung Yau, Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay, Ann. of Math. (2) 105 (1977), no. 2, 225–264. MR 437797, DOI 10.2307/1970998
- Luen-Fai Tam, Liouville properties of harmonic maps, Math. Res. Lett. 2 (1995), no. 6, 719–735. MR 1362965, DOI 10.4310/MRL.1995.v2.n6.a5
- Gang Tian and Shing-Tung Yau, Complete Kähler manifolds with zero Ricci curvature. II, Invent. Math. 106 (1991), no. 1, 27–60. MR 1123371, DOI 10.1007/BF01243902
- Shing Tung Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228. MR 431040, DOI 10.1002/cpa.3160280203
- Sai-Kee Yeung, Complete Kähler manifolds of positive Ricci curvature, Math. Z. 204 (1990), no. 2, 187–208. MR 1055985, DOI 10.1007/BF02570867
Additional Information
- Lei Ni
- Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093
- MR Author ID: 640255
- Email: lni@math.ucsd.edu
- Yuguang Shi
- Affiliation: Department of Mathematics, Peking University, Beijing, 100871, China
- Email: ygshi@math.pku.edu.cn
- Luen-Fai Tam
- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China
- MR Author ID: 170445
- Email: lftam@math.cuhk.edu.hk
- Received by editor(s): July 25, 2002
- Published electronically: December 18, 2002
- Additional Notes: The research of the first author was partially supported by NSF grant DMS-0196405 and DMS-0203023, USA
The research of the second author was partially supported by NSF of China, project 10001001
The research of the third author was partially supported by Earmarked Grant of Hong Kong #CUHK4217/99P - © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 1933-1959
- MSC (2000): Primary 32Q15
- DOI: https://doi.org/10.1090/S0002-9947-02-03242-7
- MathSciNet review: 1953533