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Moduli of Einstein–Hermitian harmonic mappings of the projective line into quadrics

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Abstract

The present article studies the class of Einstein–Hermitian harmonic maps of constant Kähler angle from the projective line into quadrics. We provide a description of their moduli spaces up to image and gauge equivalence using the language of vector bundles and representation theory. It is shown that the dimension of the moduli spaces is independent of the Einstein–Hermitian constant and rigidity of the associated real standard, and totally real maps are examined. Finally, certain classical results concerning embeddings of two-dimensional spheres into spheres are rephrased and derived in our formalism.

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Notes

  1. Although \(\varphi \) is traditionally termed the EH constant in the bibliography, we will adopt a slightly different convention by reasons that will appear obvious later (see Definition 2 and Remark 3).

References

  1. Bolton, J., Jensen, G.R., Rigoli, M., Woodward, L.M.: On conformal minimal immersions of \(S^{2}\) into \({{\mathbf{C}}} {\rm P}^{n}\), Mathematische Annalen. 279(4): 599–620 (1987/88)

  2. Calabi, E.: Isometric imbedding of complex manifolds. Ann. Math. 58, 1–23 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, B.Y., Ogiue, K.: On totally real submanifolds. Trans. Am. Math. Soc. 193, 257–266 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chern, S.S., Wolfson, J.: Minimal surfaces by moving frames. Am. J. Math. 105, 59–83 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  5. do Carmo, M.P., Wallach, N.R.: Minimal immersions of spheres into spheres. Ann. Math 93, 43–62 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  6. Eells, J., Lemaire, L.: A report on Harmonic maps. Bull. Lond. Math. Soc. 10, 1–68 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Iwanami Shoten and Princeton University, Tokyo (1987)

    Book  MATH  Google Scholar 

  8. Koga, I., Nagatomo, Y.: A study of submanifolds of the complex grassmannian manifold with parallel second fundamental form. Tokyo J. Math. 39(1), 173–185 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Macia, O., Nagatomo, Y., Takahashi, M.: Holomorphic isometric embeddings of the projective line into quadrics, Tohoku Math. J. (2) 69(4), 525–545 (2017). https://www.math.tohoku.ac.jp/tmj/Nissue.html

  10. Nagatomo, Y.: Harmonic maps into Grassmannian manifolds, arXiv: mathDG/1408. 1504

  11. Ruh, E.A., Vilms, J.: The tension field of the Gauss map. Trans. Am. Math. Soc. 149, 569–573 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  12. Takeuchi, M.: Modern Spherical Functions, Translations of Mathematical Monographs, vol. 135. American Mathematical Society, Providence (1994)

    Book  Google Scholar 

  13. Wang, J., Jiao, X.: Totally real minimal surfaces in the complex hyperquadrics. Diff. Geom. and its Appl. 31, 540–555 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first-named author would like to thank the hospitality of Meiji University where part of this work was developed. The work of the first-named author was supported by the Spanish Agency of Scienctific and Technological Research (DGICT) and FEDER project MTM20136961. The work of the second-named author was supported by JSPS KAKENHI Grant Number 17K05230.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, University of Valencia, C. Dr Moliner, 50, Burjassot, 46100, Valencia, Spain

    Oscar Macia

  2. Department of Mathematics, Meiji University, Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa, 214-8571, Japan

    Yasuyuki Nagatomo

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  1. Oscar Macia
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  2. Yasuyuki Nagatomo
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Correspondence to Oscar Macia.

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Macia, O., Nagatomo, Y. Moduli of Einstein–Hermitian harmonic mappings of the projective line into quadrics. Ann Glob Anal Geom 53, 503–520 (2018). https://doi.org/10.1007/s10455-017-9585-x

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  • DOI: https://doi.org/10.1007/s10455-017-9585-x

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