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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, 35J61, 53A30 show english descriptions Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
Semilinear elliptic equations
Conformal differential geometry
53C21 - Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
35J61 - Semilinear elliptic equations
53A30 - Conformal differential geometry
 

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