Ricci flatness of asymptotically locally Euclidean metrics

Authors:
Lei Ni, Yuguang Shi and Luen-Fai Tam

Journal:
Trans. Amer. Math. Soc. **355** (2003), 1933-1959

MSC (2000):
Primary 32Q15

DOI:
https://doi.org/10.1090/S0002-9947-02-03242-7

Published electronically:
December 18, 2002

MathSciNet review:
1953533

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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.

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Additional Information

**Lei Ni**

Affiliation:
Department of Mathematics, University of California, San Diego, La Jolla, California 92093

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

Email:
lftam@math.cuhk.edu.hk

DOI:
https://doi.org/10.1090/S0002-9947-02-03242-7

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

Article copyright:
© Copyright 2002
American Mathematical Society