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# Communications in Analysis and Geometry

## Volume 18 (2010)

### Number 3

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

Pages: 555 – 578

DOI: http://dx.doi.org/10.4310/CAG.2010.v18.n3.a5

#### Author

#### Abstract

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 $A \neq 0$.