Parabolicity, Brownian Exit Time and Properness of Solitons of the Direct and Inverse Mean Curvature Flow


We study some potential theoretic properties of homothetic solitons \(\Sigma ^n\) of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in \(\mathbb {R}^{n+m}\), we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons \(\Sigma ^n\) with \(n>2\) are self-shrinkers and that parabolic IMCF-solitons of any dimension are self-expanders. We have studied too the geometric behavior of parabolic MCF and IMCF-solitons confined in a ball, the behavior of the mean exit time function for the Brownian motion defined on \(\Sigma \) as well as a classification of properly immersed MCF-self-shrinkers with bounded second fundamental form, following the lines of Cao and Li (Calc Var 46:879–889, 2013).

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  1. 1.

    Alias, L., Mastrolia, P., Rigoli, M.: Maximum Principles and Geometric Applications Springer Monographs in Maths. Springer, Berlin (2016)

    Google Scholar 

  2. 2.

    Cao, H.-D., Li, H.: A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. 46, 879–889 (2013)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Castro, I., Lerma, A.: Hamiltonian stationary self-similar solutions for Lagrangian mean curvature flow in the complex Euclidean plane. Proc. Am. Math. Soc. 138(5), 1821–1832 (2010)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Castro, I., Lerma, A.: Lagrangian homothetic solitons for the inverse mean curvature flow. Results Math. 71, 1109–1125 (2017)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Cavalcante, M.P., Espinar, J.M.: Halfspace type theorems for self-shrinkers. Bull. Lond. Math. Soc. 48, 242–250 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Chavel, I.: Eigenvalues in Riemannian geometry. Including a chapter by Burton Randol. With an appendix by Jozef Dodziuk., Pure and Applied Mathematics 115. Academic Press Inc., Orlando, FL, (1984). xiv+362 pp

  7. 7.

    Chavel, I.: Riemannian geometry. A modern introduction. Second edition. Cambridge Studies in Advanced Mathematics, 98. Cambridge University Press, Cambridge, (2006). xvi+471 pp. ISBN: 978-0-521-61954-7

  8. 8.

    Cheng, B.Y.: Riemannian submanifolds: a survey arXiv:1307.1875v1 [math.DG] (2013)

  9. 9.

    Cheng, Xu, Zhou, Detang: Volume estimate about shrinkers. Proc. Am. Math. Soc. 141(2), 687–696 (2013)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Chern, S.S., Do Carmo, M., Kobayashi, S.: Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length. Shiing-Shen Chern Selected Papers, pp. 393–409, Berlin (1978)

  11. 11.

    Colding, T.H., Minicozzi, W.P.: Generic mean curvature flow I, generic singularities. Ann. Math. 175(2), 755–833 (2012)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Ding, Qi, Xin, Y.L.: Volume growth, eigenvalue and compactness for self-shrinkers. Asian J. Math. 17(3), 443–456 (2013)

    MathSciNet  Article  Google Scholar 

  13. 13.

    do Carmo, M.P.: Riemannian geometry. Translated from the second Portuguese edition by Francis Flaherty.. Mathematics: Theory & Applications. Birkhäuser Boston Inc., MA, (1992). xiv+300 pp. ISBN: 0-8176-3490-8

  14. 14.

    Drugan, G., Lee, H., Wheeler, G.: Solitons for the Inverse mean curvature flow. Pacific. J. Math. 284(2), 309–316 (2016)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Dynkin, E.B.: Markov Processes. Springer, Berlin (1965)

    Google Scholar 

  16. 16.

    Greene, R., Wu, H.: Function Theory on Manifolds Which Possess a Pole Lecture Notes in Math, vol. 699. Springer, Berlin (1979)

    Google Scholar 

  17. 17.

    Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. (N.S.) 36(2), 135–249 (1999)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Has’minskii, R.Z.: Probabilistic representation of the solution of some differential equations. In: Proceedings of the 6th All Union Conference on Theoritical Probability and Mathematical Statistics ((Vilnius 1960) (1960)

  19. 19.

    Hurtado, A., Palmer, V., Rosales, C.: Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights. arXiv:1805.10055 (2018)

  20. 20.

    Jost, J.: Riemannian Geometry and Geometric Analysis, 3rd edn. Springer, Berlin (2002)

    Google Scholar 

  21. 21.

    Lee, J.: Introduction to Smooth Manifolds. Universitext, Springer, Berlin (2003)

    Google Scholar 

  22. 22.

    Li, A.M., Li, J.M.: An intrinsic rigidity theorem for minimal submanifolds in a sphere. Arch. Math. 58, 582–594 (1992)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Mantegazza, C.: Lecture Notes on Mean Curvature Flow Progress in Mathematics, vol. 290. Birkhauser, Springer, Basel (2011)

    Google Scholar 

  24. 24.

    Markvorsen, S.: On the mean exit time form a minimal submanifold. J. Diff. Geom. 2, 1–8 (1989)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Markvorsen, S., Min-Oo, M.: Global Riemannian Geometry: Curvature and Topology, Advanced Courses in Mathematics CRM Barcelona. Birkhäuser, Berlin (2003)

    Google Scholar 

  26. 26.

    Markvorsen, S., Palmer, V.: Transience and capacity of minimal submanifolds. Geom. Funct. Anal. 13, 915–933 (2003)

    MathSciNet  Article  Google Scholar 

  27. 27.

    McDonald, Patrick: Exit times, moment problems and comparison theorems. Potential Anal. 38, 1365–1372 (2013)

    MathSciNet  Article  Google Scholar 

  28. 28.

    McDonald, McM Patrick, Meyers, Robert: Dirichlet spectrum and heat content. J. Funct. Anal. 200(1), 150–159 (2003)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Nadirashvili, N.: Hadamard and Calabi–Yau’s conjectures on negatively curved and minimal surfaces. Inventiones Mathematicae 126(3), 457–465 (1995)

    MathSciNet  Article  Google Scholar 

  30. 30.

    O’Neill, B.: Semi-Riemannian Geometry; With Applications to Relativity Pure and Applied Mathematics Series. Academic Press, San Diego (1983)

    Google Scholar 

  31. 31.

    Palmer, V.: On deciding whether a submanifold is parabolic of hyperbolic using its mean curvature. In: Haesen, S., Verstraelen, L. (eds.) Topics in Modern Differential Geometry Atlantis Transactions in Geometry. Atlantis Press, Atlantis (2017)

    Google Scholar 

  32. 32.

    Palmer, V.: Mean exit time from convex hypersurfaces. Proc. Am. Math. Soc. 126, 2089–2094 (1998)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Palmer, V.: Isoperimetric inequalities for extrinsic balls in minimal submanifolds and their applications. J. Lond. Math. Soc. 60(2), 607–616 (1999)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Pigola, S., Rigoli, M., Setti, A.G.: Maximum principles on Riemannian manifolds and applications. Mem. Am. Math. Soc. 174, 822 (2005)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Pigola, S., Rimoldi, M.: Complete self-shinkers confined into some regions of the space. Ann. Glob. Anal. Geom. 45, 47–65 (2014)

    Article  Google Scholar 

  36. 36.

    Rimoldi, M.: On a classification theorem for self-shrinkers. Proc. Am. Math. Soc. 124, 3605–3613 (2014)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Sakai, T.: Riemannian Geometry, Translations of Mathematical Monographs, vol. 149. American Mathematical Society, Providence (1996)

    Google Scholar 

  38. 38.

    Simon, J.: Minimal varieties in Riemannian manifolds. Ann. Math. 88, 62–105 (1968)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Smoczyk, K.: Self-shrinkers of the mean curvature flow in arbitrary codimension. Int. Math. Res. Not. 48, 2983–3004 (2005)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Takahasi, T.: Minimal immersions of Riemannian manifolds. J. Math. Soc. Jpn. 18(4), 380–385 (1966)

    MathSciNet  Article  Google Scholar 

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Correspondence to Vicente Palmer.

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Vicent Gimeno: Work partially supported by the Research Program of University Jaume I Project UJI-B2018-35, and DGI -MINECO Grant (FEDER) MTM2017-84851-C2-2-P. Vicente Palmer: Work partially supported by the Research Program of University Jaume I Project UJI-B2018-35, DGI-MINECO Grant (FEDER) MTM2017-84851-C2-2-P, and Generalitat Valenciana Grant PrometeoII/2014/064.

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Gimeno, V., Palmer, V. Parabolicity, Brownian Exit Time and Properness of Solitons of the Direct and Inverse Mean Curvature Flow. J Geom Anal 31, 579–618 (2021).

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  • Extrinsic distance
  • Parabolicity
  • Soliton
  • Self-shrinker
  • Self-expander
  • Mean exit time function
  • Laplace operator
  • Brownian motion
  • Mean curvature flow
  • Inverse mean curvature flow

Mathematics Subject Classification

  • Primary 53C21
  • 53C44
  • Secondary 53C42
  • 58J65
  • 60J65