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The curve shortening flow with density of a spherical curve in codimension two

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Abstract

In the present paper we carry out a systematic study about the flow of a spherical curve by the mean curvature flow with density in a 3-dimensional rotationally symmetric space with density \((M^3_w,\,g_w,\,\xi )\) where the density \(\xi \) decomposes as sum of a radial part \(\varphi \) and an angular part \(\psi \). We analyse how either the parabolicity or the hyperbolicity of \((M^3_w,\,g_w)\) conditions the behaviour of the flow when the solution goes to infinity.

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References

  1. Angenent, S. Parabolic equations for curves on surfaces. i. curves with p-integrable curvature. Annals of Mathematics (2)132, 3 (1990) 451–483.

  2. Angenent, S. Parabolic equations for curves on surfaces. ii. intersections, blow-up and generalized solutions. Annals of Mathematics (2) 133 1 (January 1991), 171–215.

  3. Borisenko, A., and Miquel, V. Gaussian mean curvature flow. Journal of Evolution Equations 10 (2010), 415–423.

    Article  MathSciNet  Google Scholar 

  4. Castro, I., Lerma, A. M., and Miquel, V. Evolution by mean curvature flow of lagrangian spherical surfaces in complex euclidean plane. Journal of Mathematical Analysis and Applications 462 (2018), 637–647.

    Article  MathSciNet  Google Scholar 

  5. Chen, J., and Li, J. Mean curvature flow of surface in 4-manifolds. Advances in Mathematics 163, 2 (November 2001), 287–309.

  6. Chou, K.-S., and Zhu, X.-P. The Curve Shortening Problem. Chapman and Hall/CRC, Boca Raton, FL, March 2001.

  7. Gage, M., and Hamilton, R. The heat equation shrinking convex plane curves. Journal Differential Geometry 23 (1986), 69–96.

    Article  MathSciNet  Google Scholar 

  8. Gage, M. E. Curve shortening makes convex curves circular. Inventiones mathematicae 76 (1984), 357–364.

    Article  MathSciNet  Google Scholar 

  9. Gage, M. E. Curve shortening on surfaces. Annales Scientifiques de l’École Normale Supérieure 23, 2 (1990), 229–256.

    Article  MathSciNet  Google Scholar 

  10. Grayson, M. Shortening embedded curves. Annals of Mathematics 129 (1989), 71–111.

    Article  MathSciNet  Google Scholar 

  11. Grigor’yan, A. Analytic and geometric background of recurrence and non-explosion of the brownian motion on riemannian manifolds. Bull. Amer. Math. Soc. (N.S.) 36 (1999), 135–249.

  12. Hamilton, R. Three-manifolds with positive ricci curvature. Journal Differential Geometry 17, 2 (1982), 255–306.

    Article  MathSciNet  Google Scholar 

  13. Huisken, G. Asymptotic behavior for singularities of the mean curvature flow. Journal Differential Geometry 31, 1 (1990), 285 – 299.

    Article  MathSciNet  Google Scholar 

  14. Ma, L., and Chen, D. Curve shortening in a riemannian manifold. Annali di Matematica Pura ed Applicata 186, 4 (October 2007), 663–684.

  15. Miquel, V., and Viñado-Lereu, F. The curve shortening problem associated to a density. Calculus of Variations and Partial Differential Equations 55:61, 3 (June 2016), 1–30.

  16. Miquel, V., and Viñado-Lereu, F. Type i singularities in the curve shortening flow associated to a density. The Journal of Geometric Analysis 28, 3 (July 2018), 2361–2394.

    Article  MathSciNet  Google Scholar 

  17. Oaks, J. A. Singularities and self intersections of curve evolving on surfaces. Indiana University Mathematics Journal 43, 3 (1994), 959–981.

    Article  MathSciNet  Google Scholar 

  18. O’Neill, B. Semi-Riemannian Geometry with Applications to Relativity. 1983.

  19. Schnürer, O. C., and Smoczyk, K. Evolution of hypersurfaces in central force fields. Journal für die reine und angewandte Mathematik 550 (2002), 77–95.

    MathSciNet  MATH  Google Scholar 

  20. Smoczyk, K. Symmetric hypersurfaces in riemannian manifolds contracting to lie-groups by their mean curvature. Calculus of Variations and Partial Differential Equations 4, 2 (February 1996), 155 – 170.

  21. Smoczyk, K. Mean curvature flow in higher codimension: Introduction and survey. In Global Differential Geometry., M. S. Christian Bär, Joachim Lohkamp, Ed., vol. 17 of Springer Proceedings in Mathematics. Springer, Berlin, Heidelberg, 2012, pp. 231–274.

  22. Stoker, J. Differential Geometry. Wiley-Interscience, 1969.

  23. Wang, M.-T. Mean curvature flow of surfaces in einstein four-manifolds. Journal of Differential Geometry 57, 2 (2001), 301–338.

    MathSciNet  MATH  Google Scholar 

  24. Zhou, H. Curve shortening flows in warped product manifolds. Proceedings of the American Mathematical Society 145, 10 (October 2017), 4503–4516.

  25. Zhu, X.-P. Asymptotic behavior of anisotropic curve flows. Journal Differential Geometry 48 (1998), 225–274.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author thanks Dr. Vicent Gimeno and Dr. Vicente Palmer for their support and useful comments throughout our fruitful seminars that encouraged the writing of the present manuscript. The author also thanks the referee for the detailed comments and valuable suggestions that greatly helped to improve the quality of this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, School of Technology and Experimental Sciences, Universitat Jaume I, Av. de Vicente Sos Baynat, s/n, 12071, Castelló de la Plana, Spain

    Francisco Viñado-Lereu

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  1. Francisco Viñado-Lereu
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Correspondence to Francisco Viñado-Lereu.

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Research partially supported by Universitat Jaume I research project UJI-B2018-35 and by MINECO research project MTM2017-84851-C2-2-P. The author has been supported by a postdoctoral grant from Plan de promoción de la investigación de la Universitat Jaume I del año 2018 Acción 3.2. POSDOC-A/2018/32 - grupo 041.

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Viñado-Lereu, F. The curve shortening flow with density of a spherical curve in codimension two. J. Evol. Equ. 21, 1119–1148 (2021). https://doi.org/10.1007/s00028-020-00620-y

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