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Local Isometric Imbedding of a Compact Riemann Surface with a Singular Non-CSC Extremal K\(\ddot{a}\)hler Metric into 3-Dimension Space Forms

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Abstract

On any compact Riemann surface there always exists a singular non-CSC (constant scalar curvature) extremal K\(\ddot{a}\)hler metric which is called a non-CSC HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric. In this paper, we consider the problem whether or not a non-CSC HCMU metric can be isometrically imbedded into 3-dimension space forms. This problem was first proposed by Peng and Wu in (Results Math 75:133, 2020). By Moving frames, we show that any non-CSC HCMU metric can be locally imbedded into the 3-dimension space forms. As an application, we show that any non-CSC HCMU metric can be locally imbedded into \(\mathbb {C}P^{3}\).

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Acknowledgements

The authors would like to express their deep gratitude to the referee for his/her very valuable comments on improving the whole paper.

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Authors and Affiliations

  1. School of Mathematical and Statistics, Henan University, Kaifeng, 475004, People’s Republic of China

    Zhiqiang Wei

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Yingyi Wu

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  1. Zhiqiang Wei
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  2. Yingyi Wu
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Correspondence to Zhiqiang Wei or Yingyi Wu.

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Supported by the National Natural Science Foundation of China (Grant No. 11971450).

Supported by Natural Science Foundation of Henan, No. 202300410067.

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Wei, Z., Wu, Y. Local Isometric Imbedding of a Compact Riemann Surface with a Singular Non-CSC Extremal K\(\ddot{a}\)hler Metric into 3-Dimension Space Forms. J Geom Anal 32, 27 (2022). https://doi.org/10.1007/s12220-021-00756-4

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