On the Kähler manifolds with the largest infimum of spectrum of Laplace–Beltrami operators and sharp lower bound of Ricci or holomorphic bisectional curvatures

The paper studies the extremal or rigidity problem associated to the largestinfimum of spectrum of Laplace-Beltrami operator $\Delta_g$ on Kähler manifolds ($M^n, g$) under the sharp lower bound assumption on either Ricci curvature or holomorphic bisectional curvature. The paper provides some conterexamples on those rigidity problems. In particular, we consider $D(A) = \{z\in \mathbb{C}^n : |z|^2 + \hbox{Re} \sum^n_{j=1} A_j z^2_j \lt 1\}$ a convex domain in $\mathbb{C}^n$ with $n > 1$ and $A_j \in (-1, 1)$. Assuming $g_0$ is the Kähler-Einstein metric on $D(A)$, we prove that $\lambda_1(\Delta_{g_0}) = n^2$ on $(D(A), g_0)$, but $D(A)$ is not biholomorphic to the unit ball $B_n$ when $A \neq 0$. Moreover, we prove that $\rho (z) = -e^u$ is strictly plurisubharmonic in $D(A)$where $u$ is the potential function for Kähler-Einstein metric on $D(A)$. We also construct a complete Kähler metric $g_1$ on $D(A)$ with holomorphic bisectional curvature $\mathcal{K}_{g_1} \geq -1$ and $\lambda_1(\Delta_{g_1}) = n^2$, but $D(A)$ is not biholomorphic to $B_n$ when $A \neq 0$.

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Published 1 January 2010