Potential Analysis

, Volume 24, Issue 2, pp 137–194 | Cite as

Convergence of Riemannian Manifolds and Laplace Operators, II

Dedicated to Professor Takushiro Ochiai on his sixtieth birthday
  • Atsushi Kasue


We study spectral convergence of compact Riemannian manifolds or more generally certain Dirichlet spaces, obtaining some compactness results on harmonic functions and harmonic maps.


Riemannian manifold spectral convergence Dirichlet space harmonic function harmonic map 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Carlen, E.A., Kusuoka, A. and Stroock, D.W.: ‘Upper bounds for symmetric Markov transition functions’, Ann. Inst. H. Pincaré 23 (1987), 245–287. MathSciNetGoogle Scholar
  2. 2.
    Cassels, J.W.S.: An Introduction to the Geometry of Numbers, Springer-Verlag, Berlin, 1971. Google Scholar
  3. 3.
    Chung, K.L.: Lectures from Markov Processes to Brownian Motion, Springer-Verlag, New York, 1982. Google Scholar
  4. 4.
    Dal Maso, G.: An Introduction to Γ-Convergence, Progress in Nonlinear Differential Equations and Their Applications 8, Birkhäuser, Boston, 1993. Google Scholar
  5. 5.
    Davies, E.B.: ‘Explicit constants for Gaussian upper bounds for heat kernels’, Amer. J. Math. 109 (1987), 319–334. MATHMathSciNetGoogle Scholar
  6. 6.
    Davies, E.B.: Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. Google Scholar
  7. 7.
    Eells, J. and Fuglede, B.: Harmonic Maps between Riemannian Polyhedra, Cambridge University Press, Cambridge, 2001. Google Scholar
  8. 8.
    Feyel, D. and de La Pradelle, A.: ‘Construction d'un espace harmonique de Brelot associé à un espace de Dirichlet de type local vérifiant une hypotheèse d'hypoellipticité’, Invent. Math. 44 (1978), 109–128. CrossRefMathSciNetGoogle Scholar
  9. 9.
    Fuglede, B.: ‘Hölder continuity of harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature’, Cal. Var. Partial Differential Equations 16 (2003), 375–403. MATHMathSciNetGoogle Scholar
  10. 10.
    Fuglede, B.: ‘The Dirichlet problem for harmonic maps from Riemannian polyhedra togeodesic spaces of upper bounded curvature’, Preprint Series 2001, No. 18, University of Copenhagen, Dec. 2001. Google Scholar
  11. 11.
    Fukushima, M., Oshima, Y. and Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin, 1994. Google Scholar
  12. 12.
    Jäger, W. and Kaul, H.: ‘Uniqueness and stability of harmonic maps and their Jacobi fields’, Manuscripta Math. 28 (1979), 269–291. CrossRefMathSciNetGoogle Scholar
  13. 13.
    Jost, J.: ‘Generalized Dirichlet forms and harmonic maps’, Calc. Var. Partial Differential Equations 5 (1997), 1–19. MATHMathSciNetGoogle Scholar
  14. 14.
    Kasue, A.: ‘Convergence of Riemannian manifolds and harmonic maps’, Math. Preprint Series, Osaka City University, 2000 (unpublished). Google Scholar
  15. 15.
    Kasue, A.: ‘Convergence of Riemannian manifolds and Laplace operators, I’, Ann. Inst. Fourier 52 (2002), 1219–1257. MATHMathSciNetGoogle Scholar
  16. 16.
    Kasue, A.: ‘Convergence of Riemannian manifolds and Laplace operators, III’, in preparation. Google Scholar
  17. 17.
    Kasue, A.: ‘Convergence of metric graphs and energy forms’, Preprint. Google Scholar
  18. 18.
    Kasue, A.: ‘Convergence of Riemannian vector bundles and energy forms’, in preparation. Google Scholar
  19. 19.
    Kasue, A. and Kumura, H.: ‘Spectral convergence of Riemannian manifolds’, Tōhoku Math. J. 46 (1994), 147–179; ‘Spectral convergence of Riemannian manifolds II’, Tōhoku Math. J. 48 (1996), 71–120. MathSciNetGoogle Scholar
  20. 20.
    Kasue, A., Kumura, H. and Ogura, Y.: ‘Convergence of heat kernels on a compact manifold’, Kyuushu J. Math. 51 (1997), 453–524. MathSciNetGoogle Scholar
  21. 21.
    Kuwae, K. and Shioya, T.: ‘Convergence of spectral structures: A functional analytic theory and its applications to spectral geometry’, Comm. Anal. Geom. 11 (2003), 599–673. MathSciNetGoogle Scholar
  22. 22.
    Mosco, U.: ‘Composite media and asymptotic Dirichlet forms’, J. Funct. Anal. 123 (1994), 368–421. CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Nagano, T. and Smith, B.: ‘Minimal varieties and harmonic maps in tori’, Comment. Math. Helvetici 50 (1975), 249–265. Google Scholar
  24. 24.
    Ogura, Y.: ‘Weak convergence of laws of stochastic processes on Riemannian manifolds’, Probab. Theory Related Fields 119 (2001), 529–557. CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Ogura, Y., Tomisaki, M. and Tsuchiya, M.: ‘Convergence of local type Dirichlet forms to a non-local type one’, Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), 507–556. MathSciNetGoogle Scholar
  26. 26.
    Picard, J.: ‘Smoothness of harmonic maps for hypoelliptic diffusions’, Ann. Probab. 28 (2000), 643–666. CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Picard, J.: ‘The manifold-valued Dirichlet problem for symmetric diffusions’, Potential Anal. 14 (2001), 53–72. CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Saloff-Coste, L.: ‘A note on Poincaré, Sobolev and Harnack inequality’, Duke Math. J. Int. Math. Res. Notices 2 (1992), 27–38. MATHMathSciNetGoogle Scholar
  29. 29.
    Sturm, K.T.: ‘On the geometry defined by Dirichlet forms’, in E. Bolthausen et al. (eds), Seminar on Stochastic Processes, Random Fields and Applications (Ascona), Progress in Probability 36, Birkhäuser, 1995. Google Scholar
  30. 30.
    Sturm, K.T.: ‘Analysis on local Dirichlet spaces III. The parabolic Harnack inequality’, J. Math. Pures Appl. 75 (1996), 273–297. MATHMathSciNetGoogle Scholar
  31. 31.
    Tam, L.F.: ‘Liouville properties of harmonic maps’, Math. Res. Lett. 2 (1995), 719–735. MATHMathSciNetGoogle Scholar
  32. 32.
    Yau, S.-T.: ‘Some function theoretic properties of complete Riemannian manifolds and their applications to geometry’, Indiana Math. J. 25 (1976), 659–670. MATHGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Atsushi Kasue
    • 1
  1. 1.Department of MathematicsKanazawa UniversityKanazawaJapan

Personalised recommendations