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Invariant hypersurface flows in centro-affine geometry

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Abstract

In this paper, the invariant geometric flows for hypersurfaces in centro-affine geometry are explored. We first present evolution equations of the centro-affine invariants corresponding to the geometric flows. Based on these fundamental evolution equations, we show that the centro-affine heat flow for hypersurfaces is equivalent to a system of ordinary differential equations, which can be solved explicitly. Finally, the centro-affine invariant normal flows for hypersurfaces are investigated, and two specific flows are provided to illustrate the behaviour of the flows.

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References

  1. Abresch U, Langer J. The normalized curve shortening flow and homothetic solutions. J Differential Geom, 1986, 23: 175–196

    Article  MathSciNet  MATH  Google Scholar 

  2. Altschuler S J. Singularities of the curve shrinking flow for space curves. J Differential Geom, 1991, 34: 491–514

    Article  MathSciNet  MATH  Google Scholar 

  3. Andrews B. Contraction of convex hypersurfaces by their affine normal. J Differential Geom, 1996, 43: 207–230

    Article  MathSciNet  MATH  Google Scholar 

  4. Andrews B. The affine curve-lengthening flow. J Reine Angew Math, 1999, 506: 43–83

    Article  MathSciNet  MATH  Google Scholar 

  5. Andrews B, Guan P F, Ni L. Flow by powers of the Gauss curvature. Adv Math, 2016, 299: 174–201

    Article  MathSciNet  MATH  Google Scholar 

  6. Angenent S. On the formation of singularities in the curve shortening flow. J Differential Geom, 1991, 33: 601–633

    Article  MathSciNet  MATH  Google Scholar 

  7. Angenent S, Sapiro G, Tannenbaum A. On the affine heat equation for non-convex curves. J Amer Math Soc, 1998, 11: 601–634

    Article  MathSciNet  MATH  Google Scholar 

  8. Blaschke W. Vorlesungen über Differentialgeometrie, Volume II. Berlin: Springer, 1923

    MATH  Google Scholar 

  9. Brakke K A. The Motion of a Surface by Its Mean Curvature. Mathematical Notes, vol. 20. Princeton: Princeton University Press, 1978

    MATH  Google Scholar 

  10. Cheng X X, Hu Z J, Moruz M, et al. On product affine hyperspheres in ℝn+1. Sci China Math, 2020, 63: 2055–2078

    Article  MathSciNet  MATH  Google Scholar 

  11. Chou K S, Kwong Y C. On quasilinear parabolic equations which admit global solutions for initial data with unrestricted growth. Calc Var Partial Differential Equations, 2001, 12: 281–315

    Article  MathSciNet  MATH  Google Scholar 

  12. Chou K S, Li G X. Optimal systems and invariant solutions for the curve shortening problem. Comm Anal Geom, 2002, 10: 241–274

    Article  MathSciNet  MATH  Google Scholar 

  13. Chou K S, Zhu X P. Shortening complete plane curves. J Differential Geom, 1998, 50: 471–504

    Article  MathSciNet  MATH  Google Scholar 

  14. Chou K S, Zhu X P. Anisotropic flows for convex plane curves. Duke Math J, 1999, 97: 579–619

    Article  MathSciNet  MATH  Google Scholar 

  15. Chou K S, Zhu X P. The Curve Shortening Problem. Boca Raton: Chapman & Hall/CRC, 2001

    Book  MATH  Google Scholar 

  16. Chow B. Deforming convex hypersurfaces by the nth root of the Gaussian curvature. J Differential Geom, 1985, 22: 117–138

    Article  MathSciNet  MATH  Google Scholar 

  17. Chow B. Deforming convex hypersurfaces by the square root of the scalar curvature. Invent Math, 1987, 87: 63–82

    Article  MathSciNet  MATH  Google Scholar 

  18. Daskalopoulos P, Sesum N. Ancient solutions to geometric flows. Notices Amer Math Soc, 2020, 67: 467–474

    Article  MathSciNet  MATH  Google Scholar 

  19. Evans L C. Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. Providence: Amer Math Soc, 1998

    MATH  Google Scholar 

  20. Fels M, Olver P J. Moving coframes: II. Regularization and theoretical foundations. Acta Appl Math, 1999, 55: 127–208

    Article  MathSciNet  MATH  Google Scholar 

  21. Gage M E, Hamilton R S. The heat equation shrinking convex plane curves. J Differential Geom, 1986, 23: 69–96

    Article  MathSciNet  MATH  Google Scholar 

  22. Gardner R B, Wilkens G R. The fundamental theorems of curves and hypersurfaces in centro-affine geometry. Bull Belg Math Soc Simon Stevin, 1997, 4: 379–401

    Article  MathSciNet  MATH  Google Scholar 

  23. Grayson M. The heat equation shrinks embedded plane curves to round points. J Differential Geom, 1987, 26: 285–314

    Article  MathSciNet  MATH  Google Scholar 

  24. Grayson M. Shortening embedded curves. Ann of Math (2), 1989, 129: 71–111

    Article  MathSciNet  MATH  Google Scholar 

  25. Grayson M. The shape of a figure-eight under the curve shortening flow. Invent Math, 1989, 96: 177–180

    Article  MathSciNet  MATH  Google Scholar 

  26. Hamilton R S. The Formation of Singularities in the Ricci Flow. Surveys in Differential Geometry, vol. 2. Cambridge: International Press, 1995

    MATH  Google Scholar 

  27. Huisken G. Flow by mean curvature of convex surfaces into spheres. J Differential Geom, 1984, 20: 237–266

    Article  MathSciNet  MATH  Google Scholar 

  28. Li A M, Wang C P. Canonical centroaffine hypersurfaces in ℝn+1. Results Math, 1991, 20: 660–681

    Article  MathSciNet  MATH  Google Scholar 

  29. Li X X, Zhang D. The blow-up of the conformal mean curvature flow. Sci China Math, 2020, 63: 733–754

    Article  MathSciNet  MATH  Google Scholar 

  30. Li Y. On an extension of the Hk mean curvature flow. Sci China Math, 2012, 55: 99–118

    Article  MathSciNet  Google Scholar 

  31. Loftin J, Tsui M. Ancient solutions of the affine normal flow. J Differential Geom, 2008, 78: 113–162

    Article  MathSciNet  MATH  Google Scholar 

  32. Mullins W W. Two-dimensional motion of idealized grain boundaries. J Appl Phys, 1956, 27: 900–904

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  34. Oaks J A. Singularities and self intersections of curves evolving on surfaces. Indiana Univ Math J, 1994, 43: 959–981

    Article  MathSciNet  MATH  Google Scholar 

  35. Olver P J. Invariant submanifold flows. J Phys A, 2008, 41: 344017

    Article  MathSciNet  MATH  Google Scholar 

  36. Olver P J. Moving frames and differential invariants in centro-affine geometry. Lobachevskii J Math, 2010, 31: 77–89

    Article  MathSciNet  MATH  Google Scholar 

  37. Olver P J. Modern developments in the theory and applications of moving frames. In: Impact 150: Stories of the Impact of Mathematics. London: London Mathematical Society, 2015, 14–50

    Google Scholar 

  38. Olver P J, Qu C Z, Yang Y. Feature matching and heat flow in centro-affine geometry. SIGMA Symmetry Integrability Geom Methods Appl, 2020, 16: 093

    MathSciNet  MATH  Google Scholar 

  39. Olver P J, Sapiro G, Tannenbaum A. Differential invariant signatures and flows in computer vision: A symmetry group approach. In: Geometry-Driven Diffusion in Computer Vision. Computational Imaging and Vision, vol. 1. Dordrecht: Springer, 1994, 255–306

    Chapter  Google Scholar 

  40. Olver P J, Sapiro G, Tannenbaum A. Affine invariant detection: Edge maps, anisotropic diffusion, and active contours. Acta Appl Math, 1999, 59: 45–77

    Article  MathSciNet  MATH  Google Scholar 

  41. Pekşen Ö, Khadjiev D. On invariants of curves in centro-affine geometry. J Math Kyoto Univ, 2004, 44: 603–613

    MathSciNet  MATH  Google Scholar 

  42. Sapiro G, Tannenbaum A. On affine plane curve evolution. J Funct Anal, 1994, 119: 79–120

    Article  MathSciNet  MATH  Google Scholar 

  43. Simon U. Affine differential geometry. In: Handbook of Differential Geometry, vol. 1. Amsterdam: North-Holland, 2000, 905–961

    Chapter  Google Scholar 

  44. Simon U, Schwenk-Schellschmidt A, Viesel H. Introduction to the Affine Differential Geometry of Hypersurfaces. Tokyo: Science University Tokyo Press, 1991

    MATH  Google Scholar 

  45. Tso K. Deforming a hypersurface by its Gauss-Kronecker curvature. Comm Pure Appl Math, 1985, 38: 867–882

    Article  MathSciNet  MATH  Google Scholar 

  46. Wang C P. Centroaffine minimal hypersurfaces in ℝn+1. Geom Dedicata, 1994, 51: 63–74

    Article  MathSciNet  MATH  Google Scholar 

  47. Wilkens G R. Centro-affine geometry in the plane and feedback invariants of two-state scalar control systems. In: Differential Geometry and Control. Proceedings of Symposia in Pure Mathematics, vol. 64. Providence: Amer Math Soc, 1999, 319–333

    Chapter  MATH  Google Scholar 

  48. Wo W F, Wang X L, Qu C Z. The centro-affine invariant geometric heat flow. Math Z, 2018, 288: 311–331

    Article  MathSciNet  MATH  Google Scholar 

  49. Yang Y, Yu Y, Liu H. Linear Weingarten centroaffine translation surfaces in ℝ3. J Math Anal Appl, 2011, 375: 458–466

    Article  MathSciNet  MATH  Google Scholar 

  50. Yang Y, Yu Y, Liu H. Centroaffine geometry of equiaffine rotation surfaces in ℝ3. J Math Anal Appl, 2014, 414: 46–60

    Article  MathSciNet  MATH  Google Scholar 

  51. Zhang Z H. A note on the backwards uniqueness of the mean curvature flow. Sci China Math, 2019, 62: 1793–1798

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11631007 and 11971251).

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Authors and Affiliations

  1. Department of Mathematics, Northeastern University, Shenyang, 110819, China

    Yun Yang

  2. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, China

    Changzheng Qu

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  1. Yun Yang
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  2. Changzheng Qu
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Correspondence to Changzheng Qu.

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Yang, Y., Qu, C. Invariant hypersurface flows in centro-affine geometry. Sci. China Math. 64, 1715–1734 (2021). https://doi.org/10.1007/s11425-020-1831-8

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  • DOI: https://doi.org/10.1007/s11425-020-1831-8

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