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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Prescribed scalar curvature equation on $ S^n$ in the presence of reflection or rotation symmetry


Authors: Man Chun Leung and Feng Zhou
Journal: Proc. Amer. Math. Soc. 142 (2014), 1607-1619
MSC (2010): Primary 35J60; Secondary 53C21
Published electronically: February 11, 2014
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Abstract: Using the flow equation for the conformal scalar curvature equation, we present existence theorems in cases where the prescribed function $ \mathcal {K}$ exhibits reflection or rotation symmetry (with fixed point set denoted by $ \mathcal {F}$). We also demonstrate that the ``one bubble'' condition, namely,

$\displaystyle \displaystyle {(\max _{S^n} \mathcal {K})^{\tau } \ < \ 2 \cdot (\max _{ \mathcal {F} } \mathcal {K})^{\tau }},$

cannot be totally taken away. Here $ {\tau ={1\over {2}} (n - 2).}$

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Additional Information

Man Chun Leung
Affiliation: Department of Mathematics, National University of Singapore, 10, Lower Kent Ridge Road, Singapore 119076, Republic of Singapore
Email: matlmc@nus.edu.sg

Feng Zhou
Affiliation: Department of Mathematics, National University of Singapore, 10, Lower Kent Ridge Road, Singapore 119076, Republic of Singapore
Email: zhoufeng@nus.edu.sg

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11993-9
PII: S 0002-9939(2014)11993-9
Keywords: Scalar curvature equation, geometric flow equation, blow-up
Received by editor(s): June 1, 2012
Published electronically: February 11, 2014
Communicated by: Lei Ni
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.