A characterization of the standard embeddings of ℂP 2 and Q 3

Jost Eschenburg Universit¨at Augsburg Maria Joao Ferreira Complexo Interdisciplinar da Universidade de Lisboa Renato Tribuzy Universidade Federal do Amazonas

Differential Geometry mathscidoc:1609.10163

Journal of Differential Geometry, 84, (2), 289-300, 2009
H. Hopf showed that the only constant mean curvature sphere S2 immersed in R3 is the round sphere. The K¨ahler framework is an adequate approach to generalize Hopf’ s theorem to higher dimensions. When ϕ : M → Rn is an isometric immersion from a K¨ahler manifold, the complexified second fundamental form α splits according to types. The (1, 1) part of the second fundamental form plays the role of the mean curvature for surfaces and will be called the pluri-mean curvature pmc. Therefore isometric immersions with parallel pluri-mean curvature (ppmc isometric immersions) generalize in a natural way the cmc immersions. It is a standard fact that R8 is the smallest space where CP2 can be embedded. The aim of this work is to generalize Hopf’s theorem proving in particular that the only ppmc isometric immersion from CP2 into R8 is the standard immersion.
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  title={A characterization of the standard embeddings of ℂP 2 and Q 3},
  author={Jost Eschenburg, Maria Joao Ferreira, and Renato Tribuzy},
  booktitle={Journal of Differential Geometry},
Jost Eschenburg, Maria Joao Ferreira, and Renato Tribuzy. A characterization of the standard embeddings of ℂP 2 and Q 3. 2009. Vol. 84. In Journal of Differential Geometry. pp.289-300. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911212002105770820.
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