Mathematische Annalen

, Volume 365, Issue 3–4, pp 1603–1632 | Cite as

Estimates of the first Dirichlet eigenvalue from exit time moment spectra

Article

Abstract

We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. As an application of the model space theory we prove lower and upper bounds for the first Dirichlet eigenvalues of extrinsic metric balls in submanifolds of ambient Riemannian spaces which have model space controlled curvatures. Moreover, from this general setting we thereby obtain new generalizations of the classical and celebrated results due to McKean and Cheung–Leung concerning the fundamental tones of Cartan–Hadamard manifolds and the fundamental tones of submanifolds with bounded mean curvature in hyperbolic spaces, respectively.

Mathematics Subject Classification

Primary 58C40 Secondary 53C20 

References

  1. 1.
    Bandle, C.: Isoperimetric inequalities and applications. In: Monographs and Studies in Mathematics, vol. 7. Pitman (Advanced Publishing Program), Boston (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. (2) 66(2), 499-512 (2002)Google Scholar
  3. 3.
    Barroso, C.S., Bessa, G.P.: A note on the first eigenvalue of spherically symmetric manifolds. Mat. Contemp. 30, 63-69 (2006). XIV School on Differential Geometry (Portuguese)Google Scholar
  4. 4.
    Barroso, C.S., Bessa, G.P.: Lower bounds for the first Laplacian eigenvalue of geodesic balls of spherically symmetric manifolds. Int. J. Appl. Math. Stat. 6(D06), 82-86 (2006)Google Scholar
  5. 5.
    Barta, J.: Sur le vibration fondamentale d’une membrane. C. R. Acad. Sci. 204, 472-473 (1937)MATHGoogle Scholar
  6. 6.
    Bessa, G.P., Montenegro, J.F.: Eigenvalue estimates for submanifolds with locally bounded mean curvature. Ann. Global Anal. Geom. 24(3), 279-290 (2003)Google Scholar
  7. 7.
    Betz, C., Cámera, G.A., Gzyl, H.: Bounds for the first eigenvalue of a spherical cap. Appl. Math. Optim. 10(3), 193-202 (1983)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Burchard, A., Shmuckenschläger, M.: Comparison theorems for exit times. GAFA 11, 651-692 (2002)MathSciNetGoogle Scholar
  9. 9.
    Chavel, I.: Eigenvalues in Riemannian geometry. In: Pure and Applied Mathematics, vol. 115. Academic Press, Orlando (1984) (including a chapter by Burton Randol, with an appendix by Jozef Dodziuk)Google Scholar
  10. 10.
    Chavel, I.: Isoperimetric inequalities. In: Cambridge Tracts in Mathematics, vol. 145. Cambridge University Press, Cambridge (2001). Differential geometric and analytic perspectivesGoogle Scholar
  11. 11.
    Cheng, S.Y.: Eigenfunctions and eigenvalues of Laplacian. In: Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 2, pp. 185-193. Amer. Math. Soc., Providence (1975)Google Scholar
  12. 12.
    Cheng, S.Y.: Eigenvalue comparison theorems and its geometric applications. Math. Z. 143(3), 289-297 (1975)Google Scholar
  13. 13.
    Cheung, L.-F., Leung, P.-F.: Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space. Math. Z. 236(3), 525-530 (2001)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    do Carmo, M.P., Warner, F.W.: Rigidity and convexity of hypersurfaces in spheres. J. Differ. Geom. 4, 133-144 (1970)Google Scholar
  15. 15.
    Dynkin, E.B.: Markov processes. vols. I, II. In: Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone, vol. 122. Die Grundlehren der Mathematischen Wissenschaften, Bände, vol. 121. Academic Press, New York (1965)Google Scholar
  16. 16.
    Friedland, S., Hayman, W.K.: Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions. Comment. Math. Helv. 51(2), 133-161 (1976)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gage, M.E.: Upper bounds for the first eigenvalue of the Laplace-Beltrami operator. Indiana Univ. Math. J. 29(6), 897-912 (1980)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Greene, R.E., Wu, H.: Function theory on manifolds which possess a pole. In: Lecture Notes in Mathematics, vol. 699. Springer, Berlin (1979)Google Scholar
  19. 19.
    Gray, A., Pinsky, M.: Mean exit time from a geodesic ball in Riemannian manifolds. Bull. Sci. Math. 107, 345-370 (1983)MathSciNetMATHGoogle Scholar
  20. 20.
    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)Google Scholar
  21. 21.
    Hurtado, A., Markvorsen, S., Palmer, V.: Torsional rigidity of submanifolds with controlled geometry. Math. Ann. 344(3), 511-542 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Hurtado, A., Markvorsen, S., Palmer, V.: Comparison of exit moment spectra for extrinsic metric balls. Potential Anal. 36(1), 137-153 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Jorge, L., Koutroufiotis, D.: An estimate for the curvature of bounded submanifolds. Am. J. Math. 103(4), 711-725 (1981)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Markvorsen, S.: On the mean exit time from a minimal submanifold. J. Differ. Geom. 29(1), 1-8 (1989)MathSciNetMATHGoogle Scholar
  25. 25.
    Markvorsen, S., Min-Oo, M.: Global Riemannian Geometry: Curvature and Topology. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser, Basel (2003)CrossRefMATHGoogle Scholar
  26. 26.
    Markvorsen, S., Palmer, V.: How to obtain transience from bounded radial mean curvature. Trans. Am. Math. Soc. 357(9), 3459-3479 (electronic) (2005)Google Scholar
  27. 27.
    Markvorsen, S., Palmer, V.: Torsional rigidity of minimal submanifolds. Proc. Lond. Math. Soc. (3) 93(1), 253-272 (2006)Google Scholar
  28. 28.
    Markvorsen, S., Palmer, V.: Extrinsic isoperimetric analysis of submanifolds with curvatures bounded from below. J. Geom. Anal. 20(2), 388-421 (2010)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Matsuzawa, T., Tanno, S.: Estimates of the first eigenvalue of a big cup domain of a $2$-sphere. Composit. Math. 47(1), 95-100 (1982)MathSciNetMATHGoogle Scholar
  30. 30.
    McDonald, P.: Isoperimetric conditions, Poisson problems, and diffusions in Riemannian manifolds. Potential Anal. 16(2), 115-138 (2002)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    McDonald, P.: Exit times, moment problems and comparison theorems. Potential Anal. 38, 1365-1372 (2013)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    McDonald, P., Meyers, R.: Dirichlet spectrum and heat content. J. Funct. Anal. 200(1), 150-159 (2003)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    McKean, H.P.: An upper bound to the spectrum of $\Delta $ on a manifold of negative curvature. J. Differ. Geom. 4, 359-366 (1970)MathSciNetMATHGoogle Scholar
  34. 34.
    O’Neill, B.: Semi-Riemannian geometry with applications to relativity. In: Pure and Applied Mathematics, vol. 103. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1983)Google Scholar
  35. 35.
    Palmer, V.: Mean exit time from convex hypersurfaces. Proc. Am. Math. Soc. 126(7), 2089-2094 (1998)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Petersen, P.: Riemannian geometry. In: Graduate Texts in Mathematics, vol. 171, 2nd edn. Springer, New York (2006)Google Scholar
  37. 37.
    Pinsky, M.A.: The first eigenvalue of a spherical cap. Appl. Math. Optim. 7(2), 137-139 (1981)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Pólya, G., Szegö, G.: Isoperimetric inequalities in mathematical physics. Annals of Mathematics Studies, vol. 27. Princeton University Press, Princeton (1951)Google Scholar
  39. 39.
    Pólya, G.: Torsional rigidity, principal frequency, electrostatic capacity and symmetrization. Q. Appl. Math. 6, 267-277 (1948)MathSciNetMATHGoogle Scholar
  40. 40.
    Sakai, T.: Riemannian geometry. In: Translations of Mathematical Monographs, vol. 149. American Mathematical Society, Providence (1996) (translated from the 1992 Japanese original by the author)Google Scholar
  41. 41.
    Sato, S.: Barta’s inequalities and the first eigenvalue of a cap domain of a $2$-sphere. Math. Z. 181(3), 313-318 (1982)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Savo, A.: On the lowest eigenvalue of the Hodge Laplacian on compact, negatively curved domains. Ann. Global Anal. Geom. 35(1), 39-62 (2009)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. V, 2nd edn. Publish or Perish Inc., Wilmington (1979)Google Scholar
  44. 44.
    van den Berg, M., Gilkey, P.B.: Heat content and a Hardy inequality for complete Riemannian manifolds. Bull. Lond. Math. Soc. 36(5), 577-586 (2004)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Yau, S.T.: Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold. Ann. Sci. École Norm. Sup. (4) 8(4), 487-507 (1975)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain
  2. 2.DTU Compute, MathematicsKgs. LyngbyDenmark
  3. 3.Departament de Matemàtiques-INITUniversitat Jaume ICastellóSpain

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