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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]) \[ \int _{{S^2}} {{e^u} \leq C{e^{\left \| {\nabla u} \right \|_2^2/16\pi \quad }}\forall u \in H_1^2({S^2})} {\text { with }}\int _{{S^2}} {u = 0} ,\], the best constant $C = {\text {Vol(}}{{\text {S}}^2}{\text {)}}$.

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Keywords: Gaussian curvature, semilinear elliptic equation
Article copyright: © Copyright 1986 American Mathematical Society