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On the structure of almost Einstein manifolds


Authors: Gang Tian and Bing Wang
Journal: J. Amer. Math. Soc. 28 (2015), 1169-1209
MSC (2010): Primary 53Cxx; Secondary 35Jxx
DOI: https://doi.org/10.1090/jams/834
Published electronically: June 17, 2015
MathSciNet review: 3369910
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Abstract: In this paper, we study the structure of the limit space of a sequence of almost Einstein manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such manifolds are the initial manifolds of some normalized Ricci flows whose scalar curvatures are almost constants over space-time in the $ L^1$-sense, and Ricci curvatures are bounded from below at the initial time. Under the non-collapsed condition, we show that the limit space of a sequence of almost Einstein manifolds has most properties which are known for the limit space of Einstein manifolds. As applications, we can apply our structure results to study the properties of Kähler manifolds.


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Additional Information

Gang Tian
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544; Beijing International Center for Mathematical Research, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
Email: tian@math.princeton.edu

Bing Wang
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
Email: bwang@math.wisc.edu

DOI: https://doi.org/10.1090/jams/834
Received by editor(s): February 24, 2013
Received by editor(s) in revised form: September 29, 2014
Published electronically: June 17, 2015
Additional Notes: The first author was partially supported by NSF Grants DMS-0804095, DMS-1309359 and an NSFC Grant
The second author was partially supported by NSF Grant DMS-1006518 and funds from SCGP
Article copyright: © Copyright 2015 American Mathematical Society