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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Einstein and conformally flat critical metrics of the volume functional


Authors: Pengzi Miao and Luen-Fai Tam
Journal: Trans. Amer. Math. Soc. 363 (2011), 2907-2937
MSC (2010): Primary 53C20; Secondary 58JXX
Published electronically: January 27, 2011
MathSciNet review: 2775792
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Abstract: Let $ R$ be a constant. Let $ \mathcal{M}^R_\gamma $ be the space of smooth metrics $ g$ on a given compact manifold $ \Omega^n $ ($ n\ge 3$) with smooth boundary $ \Sigma $ such that $ g $ has constant scalar curvature $ R $ and $ g\vert _{\Sigma}$ is a fixed metric $ \gamma$ on $ \Sigma$. Let $ V(g)$ be the volume of $ g\in\mathcal{M}^R_\gamma$. In this work, we classify all Einstein or conformally flat metrics which are critical points of $ V( \cdot )$ in $ \mathcal{M}^R_\gamma$.


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Additional Information

Pengzi Miao
Affiliation: School of Mathematical Sciences, Monash University, Victoria, 3800, Australia
Address at time of publication: Department of Mathematics, University of Miami, Coral Gables, Florida 33124
Email: Pengzi.Miao@sci.monash.edu.au, pengzim@math.miami.edu

Luen-Fai Tam
Affiliation: The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China
Email: lftam@math.cuhk.edu.hk

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05195-0
PII: S 0002-9947(2011)05195-0
Received by editor(s): January 7, 2009
Published electronically: January 27, 2011
Additional Notes: The first author was supported in part by Australian Research Council Discovery Grant #DP0987650
The second author was supported in part by Hong Kong RGC General Research Fund #CUHK403108
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.