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Gromov’s Tori Are Optimal

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Abstract

We give an optimal bound on normal curvatures of immersed n-torus in a Euclidean ball of large dimension.

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References

  1. Aubin, T.: Nonlinear Analysis on Manifolds. Monge–Ampère Equations. Grundlehren der Mathematischen Wissenschaften., vol. 252 (1982)

    Book  Google Scholar 

  2. Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Grundlehren der Mathematischen Wissenschaften., vol. 285 (1988)

    Book  Google Scholar 

  3. Fáry, I.: Sur certaines inégalités géométriques. Acta Sci. Math. 12, 117–124 (1950)

    MathSciNet  Google Scholar 

  4. Gromov, M.: Curvature, Kolmogorov diameter, Hilbert rational designs and overtwisted immersions (2022a). arXiv:2210.13256 [math.DG]

  5. Gromov, M.: Isometric immersions with controlled curvatures (2022b). arXiv:2212.06122v1 [math.DG]

  6. Gromov, M.: Scalar curvature, injectivity radius and immersions with small second fundamental forms (2022c). arXiv:2203.14013v2 [math.DG]

  7. Gromov, M., Lawson, B.: Spin and scalar curvature in the presence of a fundamental group. I. Ann. Math. (2) 111(2), 209–230 (1980)

    Article  MathSciNet  Google Scholar 

  8. Petrunin, A.: Polyhedral approximations of Riemannian manifolds. Turk. J. Math. 27(1), 173–187 (2003)

    ADS  MathSciNet  Google Scholar 

  9. Petrunin, A.: Normal curvature of Veronese embedding. MathOverflow. Eprint. https://mathoverflow.net/q/445819mathoverflow.net/q/445819

  10. Petrunin, A., Zamora Barrera, S.: What Is Differential Geometry: Curves and Surfaces (2022). arXiv:2012.11814 [math.HO]

    Google Scholar 

  11. Schur, A.: Über die Schwarzsche Extremaleigenschaft des Kreises unter den Kurven konstanter Krümmung. Math. Ann. 83(1–2), 143–148 (1921)

    Article  MathSciNet  Google Scholar 

  12. Tabachnikov, S.: The tale of a geometric inequality. In: MASS selecta: teaching and learning advanced undergraduate mathematics pp. 257–262 (2003)

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Acknowledgements

I want to thank Michael Gromov and Nina Lebedeva for help and encouragement. This work was done at Euler International Mathematical Institute of PDMI RAS.

Funding

It was partially supported by the National Science Foundation, grant DMS-2005279, and the Ministry of Education and Science of the Russian Federation, grant 075-15-2022-289.

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Authors and Affiliations

  1. Department of Mathematics, Penn State University, University Park, PA, 16802, USA

    Anton Petrunin

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  1. Anton Petrunin
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Correspondence to Anton Petrunin.

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Petrunin, A. Gromov’s Tori Are Optimal. Geom. Funct. Anal. 34, 202–208 (2024). https://doi.org/10.1007/s00039-024-00663-0

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