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

 

A best constant and the Gaussian curvature


Author: Chong Wei Hong
Journal: Proc. Amer. Math. Soc. 97 (1986), 737-747
MSC: Primary 58G30; Secondary 35B45, 53C20, 58E99
MathSciNet review: 845999
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Abstract: For axisymmetric $ f \in {C^\infty }({S^2})$ we find conditions to make $ f$ the scalar curvature of a metric pointwise conformal to the standard metric of $ {S^2}$. Closely related to these results, we prove that in the inequality (Moser [8])

$\displaystyle \int_{{S^2}} {{e^u} \leq C{e^{\left\Vert {\nabla u} \right\Vert _... ...16\pi \quad }}\forall u \in H_1^2({S^2})} {\text{ with }}\int_{{S^2}} {u = 0} ,$

, the best constant $ C = {\text{Vol(}}{{\text{S}}^2}{\text{)}}$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0845999-7
PII: S 0002-9939(1986)0845999-7
Keywords: Gaussian curvature, semilinear elliptic equation
Article copyright: © Copyright 1986 American Mathematical Society