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Evolving to non-round Weingarten spheres: integer linear Hopf flows

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Abstract

In the 1950’s Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper, we investigate conditions under which evolving a smooth convex rotationally symmetric sphere by a linear combination of its radii of curvature yields a Hopf sphere. When the coefficients of the flow have certain integer values, the fate of an initial sphere is entirely determined by the local geometry of its isolated umbilic points. A variety of behaviours is uncovered: convergence to round spheres and non-round Hopf spheres, as well as divergence to infinity. The critical quantity is the rate of vanishing of the astigmatism—the difference of the radii of curvature—at the isolated umbilic points. It is proven that the size of this quantity versus the coefficient in the flow function determines the fate of the evolution. The geometric setting for the equation is Radius of Curvature space, viewed as a pair of hyperbolic/AdS half-planes joined along their boundary, the umbilic horizon. A rotationally symmetric sphere determines a parameterized curve in this plane with end-points on the umbilic horizon. The slope of the curve at the umbilic horizon is linked by the Codazzi–Mainardi equations to the rate of vanishing of astigmatism, and for generic initial conditions can be used to determine the outcome of the flow. The slope can jump during the flow, and a number of examples are given: instant jumps of the initial slope, as well as umbilic circles that contract to points in finite time and ‘pop’ the slope. Finally, we present soliton-like solutions: curves that evolve under linear flows by mutual hyperbolic/AdS isometries (dilation and translation) of Radius of Curvature space. A forthcoming paper will apply these geometric ideas to non-linear curvature flows.

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Notes

  1. https://mathoverflow.net/questions/275214/is-there-a-simple-proof-of-the-following-identity-for-sum-k-m-1l-1km.

References

  1. Aleksandrov, A.D.: Uniqueness theorems for surfaces in the large I.-V. Am. Math. Soc. Transl. 21(2), 341–419 (1962)

    MathSciNet  MATH  Google Scholar 

  2. Andrews, B.: Contraction of convex hypersurfaces in Riemannian spaces. J. Differ. Geom. 39(2), 407–431 (1994)

    Article  MathSciNet  Google Scholar 

  3. Andrews, B.: Contraction of convex hypersurfaces in Euclidean space. Calc. Var. 2, 151–171 (1994)

    Article  MathSciNet  Google Scholar 

  4. Baran, H., Marvan, M.: Classification of integrable Weingarten surfaces possessing an sl(2)-valued zero curvature representation. Non-linearity 23, 2577–2597 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Chern, S.S.: Some new characterizations of the Euclidean sphere. Duke Math. J. 12, 279–290 (1945)

    Article  MathSciNet  Google Scholar 

  6. Chern, S.S.: On special W-surfaces. Proc. Am. Math. Soc. 6, 783–786 (1955)

    MathSciNet  MATH  Google Scholar 

  7. Chow, B., Liou, L.-P., Tsai, D.-H.: On the nonlinear parabolic equation \(\partial _t u=F(\triangle u+nu)\) on \(S^n\). Commun. Anal. Geom. 4, 415–434 (1996)

    Article  Google Scholar 

  8. Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32(1), 299–314 (1990)

    Article  MathSciNet  Google Scholar 

  9. Guilfoyle, B.: On isolated umbilic points. Commun. Anal. Geom. 28(8), 2005–2018 (2020)

    Article  MathSciNet  Google Scholar 

  10. Guilfoyle, B., Klingenberg, W.: A Neutral Kähler Surface with Applications in Geometric Optics, in Recent DEvelopments in Pseudo-Riemannian Geometry, pp. 149–178. European Mathematical Society Publishing House, Zurich (2008)

    Book  Google Scholar 

  11. Guilfoyle, B., Klingenberg, W.: On Weingarten surfaces in Euclidean and Lorentzian 3-space. Differ. Geom. Appl. 28, 454–468 (2010)

    Article  MathSciNet  Google Scholar 

  12. Guilfoyle, B., Klingenberg, W.: Parabolic classical curvature flows. J. Aust. Math. Soc. 104(3), 338–357 (2018)

    Article  MathSciNet  Google Scholar 

  13. Guilfoyle, B., Klingenberg, W.: A converging Lagrangian flow in the space of oriented lines. Kyushu J. Math. 70, 343–351 (2016)

    Article  MathSciNet  Google Scholar 

  14. Han, Q.: Deforming convex hypersurfaces by curvature functions. Analysis 17(2–3), 113–128 (1997)

    Article  MathSciNet  Google Scholar 

  15. Hartman, P., Wintner, A.: Umbilical points and W-surfaces. Am. J. Math. 76, 502–508 (1954)

    Article  MathSciNet  Google Scholar 

  16. Hopf, H.: Über Flächen mit einer Relation zwischen den Hauptkrümmungen. Math. Nachr. 4, 232–249 (1950–1951)

  17. Hopf, H.: Differential Geometry in the Large. Lecture Notes in Mathematics, vol. 1000. Springer, Berlin (1983)

    Book  Google Scholar 

  18. Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20, 237–266 (1984)

    Article  MathSciNet  Google Scholar 

  19. López, R.: Rotational linear Weingarten surfaces of hyperbolic type. Isr. J. Math. 167, 283–301 (2008)

    Article  MathSciNet  Google Scholar 

  20. Manganaro, N., Pavlov, M.V.: The constant astigmatism equation. New exact solution. J. Phys. A 47, 075203 (2014)

    Article  MathSciNet  Google Scholar 

  21. Smoczyk, K.: A representation formula for the inverse harmonic mean curvature flow. Elem. Math. 60(2), 57–65 (2005)

    Article  MathSciNet  Google Scholar 

  22. Urbas, J.I.E.: An expansion of convex hypersurfaces. J. Differ. Geom. 331, 91–125 (1991)

    MathSciNet  MATH  Google Scholar 

  23. Voss, K.: Über geschlossene Weingartensche Flächen. Math. Ann. 138, 42–54 (1959)

    Article  MathSciNet  Google Scholar 

  24. van-Brunt, B., Grant, K.: Potential applications of Weingarten surfaces in CAGD, Part I: Weingarten surfaces and surface shape investigation. Comput. Aided Geom. Design 13, 569–582 (1996)

    Article  MathSciNet  Google Scholar 

  25. Weingarten, J.: Uber die Oberflächen, für welche einer der beiden Hauptkrümmungshalbmesser eine Funktion des anderen ist. J. Reine Angew. Math. 62, 160–173 (1863)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of STEM, Munster Technological University, Kerry, Tralee, Co. Kerry, Ireland

    Brendan Guilfoyle

  2. Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, UK

    Wilhelm Klingenberg

Authors
  1. Brendan Guilfoyle
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  2. Wilhelm Klingenberg
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Correspondence to Brendan Guilfoyle.

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This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.

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Guilfoyle, B., Klingenberg, W. Evolving to non-round Weingarten spheres: integer linear Hopf flows. Partial Differ. Equ. Appl. 2, 72 (2021). https://doi.org/10.1007/s42985-021-00128-1

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