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# An Onofri-type Inequality on the Sphere with Two Conical Singularities

Published:2011-06-08
Printed: Sep 2012
• Chunqin Zhou,
Department of Mathematics, Shanghai Jiaotong University, Shanghai, 200240, China
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## Abstract

In this paper, we give a new proof of the Onofri-type inequality \begin{equation*} \int_S e^{2u} \,ds^2 \leq 4\pi(\beta+1) \exp \biggl\{ \frac{1}{4\pi(\beta+1)} \int_S |\nabla u|^2 \,ds^2 + \frac{1}{2\pi(\beta+1)} \int_S u \,ds^2 \biggr\} \end{equation*} on the sphere $S$ with Gaussian curvature $1$ and with conical singularities divisor $\mathcal A = \beta\cdot p_1 + \beta \cdot p_2$ for $\beta\in (-1,0)$; here $p_1$ and $p_2$ are antipodal.
 MSC Classifications: 53C21 - Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 35J61 - Semilinear elliptic equations 53A30 - Conformal differential geometry