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An Optimal Inequality on Locally Strongly Convex Centroaffine Hypersurfaces

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Abstract

In this paper, we establish a general inequality for locally strongly convex centroaffine hypersurfaces in \(\mathbb {R}^{n+1}\) involving the norm of the covariant differentiation of both the difference tensor K and the Tchebychev vector field T. Our result is optimal in that, applying our recent classification for locally strongly convex centroaffine hypersurfaces with parallel cubic form in Cheng et al. (Results Math, doi:10.1007/s00025-017-0651-2, 2017), we can completely classify the hypersurfaces which realize the equality case of the inequality.

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References

  1. Chen, B.Y.: An optimal inequality and extremal classes of affine spheres in centroaffine geometry. Geom. Dedicata 111, 187–210 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cheng, X., Hu, Z.: Classification of locally strongly convex isotropic centroaffine hypersurfaces (2015, preprint)

  3. Cheng, X., Hu, Z., Li, A.-M., Li, H.: On the isolation phenomena of Einstein manifolds—submanifolds versions, preprint, (2016)

  4. Cheng, X., Hu, Z., Moruz, M.: Classification of the locally strongly convex centroaffine hypersurfaces with parallel cubic form. Results Math. (2017). doi:10.1007/s00025-017-0651-2

  5. Cortés, V., Nardmann, M., Suhr, S.: Completeness of hyperbolic centroaffine hypersurfaces. Commun. Anal. Geom. 24, 59–92 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hu, Z., Li, H., Simon, U., Vrancken, L.: On locally strongly convex affine hypersurfaces with parallel cubic form. Part I. Differ. Geom. Appl. 27, 188–205 (2009)

    Article  MATH  Google Scholar 

  7. Hu, Z., Li, H., Vrancken, L.: Locally strongly convex affine hypersurfaces with parallel cubic form. J. Differ. Geom. 87, 239–307 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Li, A.-M., Li, H., Simon, U.: Centroaffine Bernstein problems. Differ. Geom. Appl. 20, 331–356 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, A.-M., Liu, H.L., Schwenk-Schellschmidt, A., Simon, U., Wang, C.P.: Cubic form methods and relative Tchebychev hypersurfaces. Geom. Dedicata 66, 203–221 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, A.-M., Simon, U., Zhao, G.: Hypersurfaces with prescribed affine Gauss-Kronecker curvature. Geom. Dedicata 81, 141–166 (2000)

  11. Li, A.-M., Simon, U., Zhao, G., Hu, Z. J.: Global affine differential geometry of hypersurfaces. In: de Gruyter Expositions in Mathematics 11, 2nd edn. Walter de Gruyter, Berlin (2015)

  12. Li, H., Vrancken, L.: A basic inequality and new characterization of Whitney spheres in a complex space form. Israel J. Math. 146, 223–242 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, A.-M., Wang, C.P.: Canonical centroaffine hypersurfaces in \(\mathbb{R}^{n+1}\). Results Math. 20, 660–681 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, H.L., Wang, C.P.: The centroaffine Tchebychev operator. Results Math. 27, 77–92 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, H.L., Wang, C.P.: Centroaffine surfaces with parallel traceless cubic form. Bull. Belg. Math. Soc. 4, 493–499 (1997)

    MathSciNet  MATH  Google Scholar 

  16. Montiel, S., Urbano, F.: Isotropic totally real submanifolds. Math. Z. 199, 55–60 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nomizu, K., Sasaki, T.: Affine Differential Geometry, Geometry of Affine Immersions. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  18. Simon, U., Schwenk-Schellschmidt, A., Viesel, H.: Introduction to the Affine Differential Geometry of Hypersurfaces. Science University Tokyo, Lecture Notes (1991)

  19. Szabó, Z.I.: Structure theorem on Riemannian symmetric space \(R(X, Y)\cdot R=0\). J. Diff. Geom. 17, 531–582 (1982)

    Article  MATH  Google Scholar 

  20. Wang, C.P.: Centroaffine minimal hypersurfaces in \(\mathbb{R}^{n+1}\). Geom. Dedicata 51, 63–74 (1994)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like express their thanks to Professors H. Li, U. Simon, and L. Vrancken for their many aspects of help for this paper. As a matter of fact, our result Theorem 1.1 could be regarded as an affine differential geometric counterpart of the main result in [12], where Li and Vrancken study a basic inequality for Lagrangian submanifolds in complex space forms and as its direct consequence they obtain a new characterization of the Whitney spheres. This project was supported by Grants of NSFC-11371330.

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  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, People’s Republic of China

    Xiuxiu Cheng & Zejun Hu

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  1. Xiuxiu Cheng
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  2. Zejun Hu
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Correspondence to Zejun Hu.

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Cheng, X., Hu, Z. An Optimal Inequality on Locally Strongly Convex Centroaffine Hypersurfaces. J Geom Anal 28, 643–655 (2018). https://doi.org/10.1007/s12220-017-9836-x

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  • DOI: https://doi.org/10.1007/s12220-017-9836-x

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