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An Onofri-type Inequality on the Sphere with Two Conical Singularities |
Canad. Math. Bull. 55(2012), 663-672
Published:2011-06-08
Printed: Sep 2012
Printed: Sep 2012
- Chunqin Zhou,
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 |