Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ahlfors L.V., Sur le type d’une surface de Riemann, C.R. Acad. Sci. Paris, 201 (1935) 30–32.
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.
Cheng S.Y., Yau S.-T., Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., 28 (1975) 333–354.
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.
Federer H., Curvature measures, Trans. Amer. Math. Soc, 93 (1959) no.3, 418–491.
Federer H., Fleming W.H., Normal and integral currents, Ann. Math., 72 (1960) 458–520.
FernÁndez, J.L., On the existence of Green’s function on Riemannian manifolds, Proc. Ams, 96 (1986) 284–286.
Greene R., Wu W., Function theory of manifolds which possess a pole, Lecture Notes Math 699, Springer, 1979.
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.
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.
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.
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.
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.
Holopainen I., Positive solutions of quasilinear elliptic equations on Riemannian manifolds, Proc. London Math. Soc. (3), 65 (1992) 651–672.
Holopainen I., Volume growth, Green’s functions and parabolicity of ends, to appear in Duke Math. J.
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.
Keselman V.M., Zorich V.A., On conformai type of a Riemannian manifold, Funct. Anal. Appl., 30 (1996) 106–117.
Kronrod A.S., On functions of two variables, (in Russian) Uspechi Matem. Nauk, 5 (1950) no.1, 24–134.
Lyons T., Sullivan D., Function theory, random paths and covering spaces, J. Diff. Geom., 19 (1984) 299–323.
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.
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.
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.
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.
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.
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.
Maz’ya V.G., Sobolev spaces, (in Russian) Izdat. Leningrad Gos. Univ. Leningrad, 1985. Engl, transl. Springer-Verlag, 1985.
Maz’ya V.G., Classes of domains, measures and capacities in the theory of differ-entiable functions, Encyclopaedia Math. Sci. 26, Springer, Berlin, 1991.
Nevanlinna R., Ein Satz über offene Riemannsche Flächen, Ann. Acad. Sci. Fenn. Part A., 54 (1940) 1–18.
Pólya G., Szegö, Isoperimetric inequalities in mathematical physics, Princeton University Press, Princeton, 1951.
Sturm K-Th., Sharp estimates for capacities and applications to symmetrical diffusions, Probability theory and related fields, 103 (1995) no.1, 73–89.
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Basel AG
About this paper
Cite this paper
Grigor’yan, A. (1999). Isoperimetric inequalities and capacities on Riemannian manifolds. In: Rossmann, J., Takáč, P., Wildenhain, G. (eds) The Maz’ya Anniversary Collection. Operator Theory: Advances and Applications, vol 109. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8675-8_9
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8675-8_9
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9726-6
Online ISBN: 978-3-0348-8675-8
eBook Packages: Springer Book Archive