Potential Analysis

, Volume 36, Issue 1, pp 137–153 | Cite as

Comparison of Exit Moment Spectra for Extrinsic Metric Balls

  • Ana Hurtado
  • Steen Markvorsen
  • Vicente Palmer


We prove explicit upper and lower bounds for the L 1-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds P m in ambient Riemannian spaces N n . We assume that P and N both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as viewed from a pole in N. The bounds for the exit moment spectra are given in terms of the corresponding spectra for geodesic metric balls in suitably warped product model spaces. The bounds are sharp in the sense that equalities are obtained in characteristic cases. As a corollary we also obtain new intrinsic comparison results for the exit time spectra for metric balls in the ambient manifolds N n themselves.


Riemannian submanifolds Extrinsic balls Torsional rigidity L1-moment spectra Exit time Isoperimetric inequalities 

Mathematics Subject Classifications (2010)

Primary 53C42 58J65 35J25 60J65 


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  1. 1.
    Bandle, C.: Isoperimetric Inequalities and Applications. Pitman Publishing Inc. (1980)Google Scholar
  2. 2.
    Bañuelos, R., van den Berg, M., Carroll, T.: Torsional rigidity and expected lifetime of Brownian motion. J. Lond. Math. Soc. 66(2), 499–512 (2002)zbMATHCrossRefGoogle Scholar
  3. 3.
    Chavel, I.: Eigenvalues in Riemannian Geometry. Academic Press (1984)Google Scholar
  4. 4.
    Chavel, I.: Isoperimetric Inequalities. Differential Geometric and Analytic Perspectives. Cambridge Tracts in Mathematics, vol. 145. Cambridge University Press (2001)Google Scholar
  5. 5.
    Cheng, S.Y., Li, P., Yau, S.T.: Heat equations on minimal submanifolds and their applications. Am. J. Math. 106, 1033–1065 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    DoCarmo, M.P., Warner, F.W.: Rigidity and convexity of hypersurfaces in spheres. J. Differ. Geom. 4, 133–144 (1970)MathSciNetGoogle Scholar
  7. 7.
    Dynkin, E.B.: Markov Processes. Springer (1965)Google Scholar
  8. 8.
    Greene, R., Wu, H.: Function Theory on Manifolds which Possess a Pole. Lecture Notes in Math., vol. 699. Springer, Berlin and New York (1979)zbMATHGoogle Scholar
  9. 9.
    Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36, 135–249 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Has’minskii, R.Z.: Probabilistic representation of the solution of some differential equations. In: Proc. 6th All Union Conf. on Theor. Probability and Math. Statist. (Vilnius 1960) (1960)Google Scholar
  11. 11.
    Hurtado, A., Markvorsen, S., Palmer, V.: Torsional rigidity of submanifolds with controlled geometry. Math. Ann. 344, 511–542 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Jorge, L.P., Koutroufiotis, D.: An estimate for the curvature of bounded submanifolds. Am. J. Math. 103, 711–725 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Kinateder, K.K.J., McDonald, P.: Variational principles for average exit time moments for diffusions in Euclidean space. Proc. Am. Math. Soc. 127, 2767–2772 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Kinateder, K.K.J., McDonald, P., Miller, D.: Exit time moments, boundary value problems, and the geometry of domains in Euclidean space. Probab. Theory Relat. Fields 111, 469–487 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Markvorsen, S.: On the mean exit time from a minimal submanifold. J. Differ. Geom. 29, 1–8 (1989)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Markvorsen, S.: On the heat kernel comparison theorems for minimal submanifolds. Proc. Am. Math. Soc. 97, 479–482 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Markvorsen, S., Min-Oo, M.: Global Riemannian Geometry: Curvature and Topology. Advanced Courses in Mathematics CRM Barcelona, Birkhäuser, Berlin (2003)zbMATHCrossRefGoogle Scholar
  18. 18.
    Markvorsen, S., Palmer, V.: Generalized isoperimetric inequalities for extrinsic balls in minimal submanifolds. J. Reine Angew. Math. 551, 101–121 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Markvorsen, S., Palmer, V.: Transience and capacity of minimal submanifolds. GAFA, Geom. Funct. Anal. 13, 915–933 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Markvorsen, S., Palmer, V.: How to obtain transience from bounded radial mean curvature. Trans. Am. Math. Soc. 357, 3459–3479 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Markvorsen, S., Palmer, V.: Torsional rigidity of minimal submanifolds. Proc. Lond. Math. Soc. 93, 253–272 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Markvorsen, S., Palmer, V.: Extrinsic isoperimetric analysis on submanifolds with curvatures bounded from below. J. Geom. Anal. 20, 388–421 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    McDonald, P.: Isoperimetric conditions, Poisson problems, and diffusions in Riemannian manifolds. Potential Anal. 16, 115–138 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Milnor, J.: Morse Theory. Annals of Mathematics Studies, Number 51. Princeton University Press (1963)Google Scholar
  25. 25.
    O’Neill, B.: Semi-Riemannian Geometry; with Applications to Relativity. Academic Press (1983)Google Scholar
  26. 26.
    Palmer, V.: Mean exit time from convex hypersurfaces. Proc. Am. Math. Soc. 126, 2089–2094 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Palmer, V.: Isoperimetric inequalities for extrinsic balls in minimal submanifolds and their applications. J. Lond. Math. Soc. 60(2), 607–616 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics. Princeton University Press (1951)Google Scholar
  29. 29.
    Pólya, G.: Torsional rigidity, principal frequency, electrostatic capacity and symmetrization. Q. Appl. Math. 6, 267–277 (1948)zbMATHGoogle Scholar
  30. 30.
    Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs, vol. 149. American Mathematical Society (1996)Google Scholar
  31. 31.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry. Publish or Perish Inc., Houston (1979)Google Scholar
  32. 32.
    van den Berg, M., Gilkey, P.B.: Heat content and Hardy inequality for complete Riemannian manifolds. Bull. Lond. Math. Soc. 36, 577–586 (2004)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain
  2. 2.Department of MathematicsTechnical University of DenmarkLyngbyDenmark
  3. 3.Departament de Matemàtiques-Institute of New Imaging TechnologiesUniversitat Jaume ICastellónSpain

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