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The holomorphic Gauss parametrization

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Abstract

We give a local parametric description of all complex hypersurfaces in \({\mathbb{C}^{n+1}}\) and in complex projective space \({\mathbb{CP}^{n+1}}\) with constant index of relative nullity, together with applications. This is a complex analogue to the parametrization for real hypersurfaces in Euclidean space known as the Gauss parametrization.

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References

  1. Abe K.: A complex analogue of Hartman–Nirenberg cylinder theorem. J. Differ. Geom. 7, 453–460 (1972)

    MATH  Google Scholar 

  2. Abe K.: Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions. Tôhoku Math. J. 25, 425–444 (1973)

    Article  MATH  Google Scholar 

  3. Dajczer M., Florit L.: On conformally flat submanifolds. Comm. Anal. Geom. 4, 261–284 (1996)

    MATH  MathSciNet  Google Scholar 

  4. Dajczer M., Florit L., Tojeiro R.: On deformable hypersurfaces in space forms. Ann. Mat. Pura Appl. 174, 361–390 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dajczer M., Gromoll D.: Gauss parametrizations and rigidity aspects of submanifolds. J. Differ. Geom. 22, 1–12 (1985)

    MATH  MathSciNet  Google Scholar 

  6. Dajczer M., Gromoll D.: Real Kaehler submanifolds and uniqueness of the Gauss map. J. Differ. Geom. 22, 13–28 (1985)

    MATH  MathSciNet  Google Scholar 

  7. Dajczer M., Gromoll D.: Euclidean hypersurfaces with isometric Gauss maps. Math. Z. 191, 201–205 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dajczer M., Rodrguez L.: On isometric immersions into complex space forms. Math. Ann. 299, 223–230 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dajczer M., Tenenblat K.: Rigidity for complete Weingarten hypersurfaces. Trans. AMS 312, 129–140 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dajczer M., Tojeiro R.: All superconformal surfaces in R  4 in terms of minimal surfaces. Math. Z. 261, 869–890 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dajczer M., Tojeiro R.: Hypersurfaces with a constant support function in spaces of constant sectional curvature. Arch. Math. 60, 296–299 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  12. Nomizu K., Smith B.: Differential geometry of complex hypersurfaces, II. J. Math. Soc. Japan 20, 498–521 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  13. Sbrana V.: Sulla varietá ad n−1 dimensioni deformabili nello spazio euclideo ad n dimensioni. Rend. Circ. Mat. Palermo 27, 1–45 (1909)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. IMPA, Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, Brazil

    Marcos Dajczer & Luis A. Florit

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  1. Marcos Dajczer
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  2. Luis A. Florit
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Correspondence to Marcos Dajczer.

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Dajczer, M., Florit, L.A. The holomorphic Gauss parametrization. manuscripta math. 129, 127–135 (2009). https://doi.org/10.1007/s00229-009-0260-9

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  • DOI: https://doi.org/10.1007/s00229-009-0260-9

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