On the lower bounds of the curvatures in a bounded domain

Abstract

Let $K_\mathcal{D} (z,\bar z)$ be the Bergman kernel of a bounded domain $\mathcal{D}$ in ℂⁿ and Sect _D (z, ξ) and Ricci _D (z, ξ) be the holomorphic sectional curvature and Ricci curvature of the Bergman metric $ds^2 = T_{\alpha \bar \beta }^\mathcal{D} (z,\bar z)dz^\alpha d\bar z^\beta$ respectively at the point $z \in \mathcal{D}$ with tangent vector ξ. It is proved by constructing suitable minimal functions that

$$Sect_\mathcal{D} (z,\xi ) \geqslant 2 - 2\left( {\frac{{n + 2}} {{n + 1}}} \right)\left[ {\frac{{K_{\mathcal{D}_1 } (z,\bar z)T_{\alpha \bar \beta }^{\mathcal{D}_1 } (z,\bar z)\xi ^\alpha \overline {\xi ^\beta } }} {{K_{\mathcal{D}_2 } (z,\bar z)T_{\lambda \bar \mu }^{\mathcal{D}_2 } (z,\bar z)\xi ^\lambda \overline {\xi ^\mu } }}} \right]^2$$

and

$$Ricci_\mathcal{D} (z,\xi ) \geqslant n + 1 - \left( {n + 2} \right)\left[ {\frac{{K_{\mathcal{D}_1 } (z,\bar z)T^{\mathcal{D}_1 } (z,\bar z)\xi ^\alpha \overline \xi ^\beta }} {{K_{\mathcal{D}_2 } (z,\bar z)T_{\lambda \bar \mu }^{\mathcal{D}_2 } (z,\bar z)\xi ^\lambda \overline \xi ^\mu }}} \right],$$

where $z \in \mathcal{D}_1 \subset \mathcal{D}_2$, $\mathcal{D}_1$ is a ball contained in $\mathcal{D}$ and $\mathcal{D}_2$ is a ball containing $\mathcal{D}$.