The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3725–3746 | Cite as

Shrinking Doughnuts via Variational Methods

  • Gregory Drugan
  • Xuan Hien Nguyen


We use variational methods and a modified curvature flow to give an alternative proof of the existence of an embedded topological \(\mathbb {S}^1 \times \mathbb {S}^{n-1}\) self-shrinking hypersurface under mean curvature flow. As a consequence of the proof, we establish an upper bound for the weighted energy of our shrinking doughnuts.


Mean curvature flow Self-shrinking Singularities Geodesics 

Mathematics Subject Classification




The authors would like to thank Stephen Kleene and Sigurd Angenent. Stephen Kleene introduced the authors in the hope to solve a related problem. That problem is still open and morphed into this one. Sigurd Angenent generously shared his expertise on the existence, properties, and convergence of parabolic flows.


  1. 1.
    Angenent, S.: Parabolic equations for curves on surfaces. I. Curves with \(p\)-integrable curvature. Ann. Math. (2) 132, 451–483 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Angenent, S.: Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions. Ann. Math. (2) 133, 171–215 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Angenent, S.B.: Shrinking doughnuts. In: Angenent, S.B. (ed.) Nonlinear Diffusion Equations and Their Equilibrium States, 3 (Gregynog, 1989), vol. 7, pp. 21–38. Birkhäuser, Boston, MA (1992)CrossRefGoogle Scholar
  4. 4.
    Drugan, G.: An immersed \(S^2\) self-shrinker. Trans. Am. Math. Soc. 367, 3139–3159 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Drugan, G., Kleene, S.J.: Immersed self-shrinkers. Trans. Am. Math. Soc. 369, 7213–7250 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gage, M.E.: Deforming curves on convex surfaces to simple closed geodesics. Indiana Univ. Math. J. 39, 1037–1059 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Oaks, J.A.: Singularities and self-intersections of curves evolving on surfaces. Indiana Univ. Math. J. 43, 959–981 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Poincaré, H.: Sur les lignes géodésiques des surfaces convexes. Trans. Am. Math. Soc. 6, 237–274 (1905)CrossRefzbMATHGoogle Scholar
  10. 10.
    Rudin, W.: Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics, 3rd edn. McGraw-Hill Book Co., New York (1976)Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Oregon Episcopal SchoolPortlandUSA
  2. 2.Department of MathematicsIowa State UniversityAmesUSA

Personalised recommendations