Isoperimetric inequalities and capacities on Riemannian manifolds

  • Alexander Grigor’yan
Part of the Operator Theory: Advances and Applications book series (OT, volume 109)


We discuss extensions of some results of V.G.Maz’ya to Riemannian manifolds. His proofs of the relationships between capacities, isoperimetric inequalities and Sobolev inequalities did not use specific properties of the Euclidean space. His method, transplanted to manifolds, gives a unified approach to such results as parabolicity criteria, eigenvalues estimates, heat kernel estimates, etc.


Riemannian Manifold Heat Kernel Sobolev Inequality Isoperimetric Inequality Geodesic Ball 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ahlfors L.V., Sur le type d’une surface de Riemann, C.R. Acad. Sci. Paris, 201 (1935) 30–32.Google Scholar
  2. [2]
    Cheeger J., A lower bound for the smallest eigenvalue of the Laplacian, in: Problems in Analysis: A Symposium in honor of Salomon Bochner, Princeton University Press. Princeton, 1970. 195–199.Google Scholar
  3. [3]
    Cheng S.Y., Yau S.-T., Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., 28 (1975) 333–354.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Coulhon T., Grigor’yan A., On-diagonal lower bounds for heat kernels on non-compact manifolds and Markov chains, Duke Math. J., 89 no.l, (1997) 133–199.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Federer H., Curvature measures, Trans. Amer. Math. Soc, 93 (1959) no.3, 418–491.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Federer H., Fleming W.H., Normal and integral currents, Ann. Math., 72 (1960) 458–520.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    FernÁndez, J.L., On the existence of Green’s function on Riemannian manifolds, Proc. Ams, 96 (1986) 284–286.zbMATHCrossRefGoogle Scholar
  8. [8]
    Greene R., Wu W., Function theory of manifolds which possess a pole, Lecture Notes Math 699, Springer, 1979.Google Scholar
  9. [9]
    Grigor’yan A., On the existence of a Green function on a manifold, (in Russian) Uspekhi Matem. Nauk, 38 (1983) no.1, 161–162. Engl, transl. Russian Math. Surveys, 38 (1983) no.l, 190-191.MathSciNetGoogle Scholar
  10. [10]
    Grigor’yan A., On the existence of positive fundamental solution of the Laplace equation on Riemannian manifolds, (in Russian) Matem. Sbornik, 128 (1985) no.3, 354–363. Engl, transl. Math. USSR Sb., 56 (1987) 349-358.MathSciNetGoogle Scholar
  11. [11]
    Grigor’yan A., Heat kernel on a non-compact Riemannian manifold, in: 1993 Summer Research Institute on Stochastic Analysis, éd. M.Pinsky et al., Proceedings of Symposia in Pure Mathematics, 57 (1994) 239–263.Google Scholar
  12. [12]
    Grigor’yan A., Estimates of heat kernels on Riemannian manifolds, to appear in Proceedings of the Instructional Conference on Spectral Theory and Geometry, Edinburgh, April 1998.Google Scholar
  13. [13]
    Grigor’yan A., Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, to appear in Bull. Amer. Math. Soc.Google Scholar
  14. [14]
    Holopainen I., Positive solutions of quasilinear elliptic equations on Riemannian manifolds, Proc. London Math. Soc. (3), 65 (1992) 651–672.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Holopainen I., Volume growth, Green’s functions and parabolicity of ends, to appear in Duke Math. J.Google Scholar
  16. [16]
    Karp L., Subharmonic functions, harmonic mappings and isometric immersions, in: Seminar on Differential Geometry, ed. S.T.Yau, Ann. Math. Stud. 102, Pince-ton, 1982.Google Scholar
  17. [17]
    Keselman V.M., Zorich V.A., On conformai type of a Riemannian manifold, Funct. Anal. Appl., 30 (1996) 106–117.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Kronrod A.S., On functions of two variables, (in Russian) Uspechi Matem. Nauk, 5 (1950) no.1, 24–134.MathSciNetzbMATHGoogle Scholar
  19. [19]
    Lyons T., Sullivan D., Function theory, random paths and covering spaces, J. Diff. Geom., 19 (1984) 299–323.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Maz’ya V.G., Classes of domains and embedding theorems for functional spaces, (in Russian) Dokl. Acad. Nauk Sssr, 133 no.3, (1960) 527–530. Engl, transl. Soviet Math. Dokl., vn1 (1961) 882-885.Google Scholar
  21. [21]
    Maz’ya V.G., The p-conductivity and theorems on imbedding certain function spaces into C-space, (in Russian) Dokl. Acad. Nauk SSSR, 140 (1961) 299–302. Engl, transl. Soviet Math. Dokl., 2 (1961) 1200-1203.Google Scholar
  22. [22]
    Maz’ya V.G., The negative spectrum of the n-dimensionai Schrödinger operator, (in Russian) Dokl. Acad. Nauk SSSR, 144 no.4, (1962) 721–722. Engl, transl. Soviet Math. Dokl., 3 (1962) 808-810.Google Scholar
  23. [23]
    Maz’ya V.G., On the theory of the n-dimensional Schrödinger operator, (in Russian) Izv. Acad. Nauk SSSR, Ser. Mat., 28 (1964) 1145–1172.Google Scholar
  24. [24]
    Maz’ya V.G., Embedding theorems and their applications, (in Russian) in: Trudy Simposiuma po teoremam vlozheniya, Baku 1966 god, ed. L.D. Kudryavzev, Nauka, Moskow, 1970.Google Scholar
  25. [25]
    Maz’ya V.G., On certain intergal inequalities for functions of many variables, (in Russian) Problemy Matematicheskogo Analiza, Leningrad. Univer., 3 (1972) 33–68. Engl, transl. J. Soviet Math., 1 (1973) 205-234.Google Scholar
  26. [26]
    Maz’ya V.G., Sobolev spaces, (in Russian) Izdat. Leningrad Gos. Univ. Leningrad, 1985. Engl, transl. Springer-Verlag, 1985.Google Scholar
  27. [27]
    Maz’ya V.G., Classes of domains, measures and capacities in the theory of differ-entiable functions, Encyclopaedia Math. Sci. 26, Springer, Berlin, 1991.Google Scholar
  28. [28]
    Nevanlinna R., Ein Satz über offene Riemannsche Flächen, Ann. Acad. Sci. Fenn. Part A., 54 (1940) 1–18.Google Scholar
  29. [29]
    Pólya G., Szegö, Isoperimetric inequalities in mathematical physics, Princeton University Press, Princeton, 1951.zbMATHGoogle Scholar
  30. [30]
    Sturm K-Th., Sharp estimates for capacities and applications to symmetrical diffusions, Probability theory and related fields, 103 (1995) no.1, 73–89.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    Varopoulos N.Th., Potential theory and diffusion of Riemannian manifolds, in: Conference on Harmonie Analysis in honor of Antoni Zygmund. Vol I, II, Wadsworth Math. Ser., Wadsworth, Belmont, Calif., 1983. 821–837.MathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 1999

Authors and Affiliations

  • Alexander Grigor’yan
    • 1
  1. 1.Department of MathematicsImperial CollegeLondonUK

Personalised recommendations