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A liouville theorem on the PDE \(\det (f_{i\bar{\jmath }})=1\)

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Abstract

Let f be a smooth plurisubharmonic function which solves

$$\begin{aligned} \det (f_{i\bar{\jmath }})=1\;\;\;\;\;\;\text{ on } \;\;\;\mathbb C^n. \end{aligned}$$

Suppose that the metric \(\omega _{f}=\sqrt{-1}f_{i\bar{\jmath }}dz_{i}\wedge d\bar{z}_{j}\) is complete and f satisfies the growth condition

$$\begin{aligned} \mathsf N_{0}^{-1}(1+|z|^2)\le f\le \mathsf N_{0}(1+ |z|^2),\;\;\;\; \text{ as }\;\;\; |z|\rightarrow \infty , \end{aligned}$$

for some \(\mathsf N_{0}>0,\) then f is a quadratic polynomial.

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Acknowledgements

We would like to thank Max-Planck-Institut für Mathematik in den Naturwissenschaften, especially Professor Jürgen Jost and Professor Xianqing Li-Jost, for their great hospitality.

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  1. Department of Mathematics, Sichuan University, Chengdu, 610064, China

    An-Min Li & Li Sheng

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  1. An-Min Li
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  2. Li Sheng
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Correspondence to Li Sheng.

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Dedicated to Udo Simon for his 80th Birthday.

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Li acknowledges the support of NSFC Grant NSFC11890663, 1196131001. Sheng acknowledges the support of NSFC Grant NSFC11871352, 1196131001.

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Li, AM., Sheng, L. A liouville theorem on the PDE \(\det (f_{i\bar{\jmath }})=1\). Math. Z. 297, 1623–1632 (2021). https://doi.org/10.1007/s00209-020-02571-z

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