General information
Latest issue
Forthcoming papers
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Mat. Sb.:

Personal entry:
Save password
Forgotten password?

Mat. Sb. (N.S.), 1985, Volume 128(170), Number 3(11), Pages 354–363 (Mi msb2164)  

This article is cited in 63 scientific papers (total in 63 papers)

On the existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds

A. A. Grigor'yan

Abstract: A Riemannian manifold is said to be parabolic if there does no exist a positive fundamental solution of the Laplace equation on it. The purpose of this article is to obtain geometric conditions, both necessary and sufficient, for a manifold to be parabolic.
Bibliography: 11 titles.

Full text: PDF file (480 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1987, 56:2, 349–358

Bibliographic databases:

UDC: 514.7
MSC: Primary 35A08, 35J05, 53C25; Secondary 31A15, 34B27
Received: 05.07.1984

Citation: A. A. Grigor'yan, “On the existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds”, Mat. Sb. (N.S.), 128(170):3(11) (1985), 354–363; Math. USSR-Sb., 56:2 (1987), 349–358

Citation in format AMSBIB
\by A.~A.~Grigor'yan
\paper On the existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds
\jour Mat. Sb. (N.S.)
\yr 1985
\vol 128(170)
\issue 3(11)
\pages 354--363
\jour Math. USSR-Sb.
\yr 1987
\vol 56
\issue 2
\pages 349--358

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. Ya. Lin, “Liouville coverings of complex spaces, and amenable groups”, Math. USSR-Sb., 60:1 (1988), 197–216  mathnet  crossref  mathscinet  zmath
    2. A. A. Grigor'yan, “On Liouville theorems for harmonic functions with finite Dirichlet integral”, Math. USSR-Sb., 60:2 (1988), 485–504  mathnet  crossref  mathscinet  zmath
    3. A. A. Grigor'yan, “Stochastically complete manifolds and summable harmonic functions”, Math. USSR-Izv., 33:2 (1989), 425–432  mathnet  crossref  mathscinet  zmath
    4. Landis E., “Application of the Potential-Theory to the Study of Qualitative Properties of Solutions of the Elliptic and Parabolic Equations”, Lect. Notes Math., 1344 (1988), 133–153  crossref  mathscinet  zmath  isi
    5. V. M. Miklyukov, V. G. Tkachev, “Some properties of the tubular minimal surfaces of arbitrary codimension”, Math. USSR-Sb., 68:1 (1991), 133–150  mathnet  crossref  mathscinet  zmath  isi
    6. Chavel I., Feldman E., “Isoperimetric Constants, the Geometry of Ends, and Large Time Heat Diffusion in Riemannian-Manifolds”, Proc. London Math. Soc., 62:2 (1991), 427–448  crossref  mathscinet  zmath  isi
    7. Thomassen C., “Isoperimetric-Inequalities and Transient Random-Walks on Graphs”, Ann. Probab., 20:3 (1992), 1592–1600  crossref  mathscinet  zmath  isi
    8. Markvorsen S., Mcguinness S., Thomassen C., “Transient Random-Walks on Graphs and Metric-Spaces with Applications to Hyperbolic Surfaces”, Proc. London Math. Soc., 64:1 (1992), 1–20  crossref  mathscinet  zmath  isi
    9. Aslanov G., “On the Neymanns Problem for 2nd-Order Differential-Equations in Unbounded Strip”, Dokl. Akad. Nauk, 330:2 (1993), 133–136  mathnet  mathscinet  zmath  isi
    10. Aslanov G., “On the Neumann Problem for 2nd-Order Elliptic-Equations in Unbounded-Domains”, Differ. Equ., 30:2 (1994), 302–305  mathnet  mathscinet  zmath  isi
    11. Lin V., Pinchover Y., “Manifolds with Group-Actions and Elliptic-Operators”, Mem. Am. Math. Soc., 112:540 (1994), 1–78  mathscinet  adsnasa  isi
    12. Zheng-Xu He, O. Schramm, “Hyperbolic and parabolic packings”, Discrete Comput Geom, 14:1 (1995), 123  crossref  mathscinet  zmath
    13. Sturm K., “Sharp Estimates for Capacities and Applications to Symmetrical Diffusions”, Probab. Theory Relat. Field, 103:1 (1995), 73–89  crossref  mathscinet  zmath  isi
    14. Konkov A., “Maximum Principle for Elliptic-Equations on Riemannian-Manifolds”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1995, no. 4, 10–14  mathscinet  zmath  isi
    15. Konkov A., “On Spaces of Solutions to Elliptic Equations on Riemannian Manifolds”, Differ. Equ., 31:5 (1995), 745–752  mathnet  mathscinet  isi
    16. Li P., Tam L., “Greens-Functions, Harmonic-Functions, and Volume Comparison”, J. Differ. Geom., 41:2 (1995), 277–318  mathscinet  zmath  isi
    17. V. A. Zorich, V. M. Kesel'man, “On the Conformal Type of a Riemannian Manifold”, Funct. Anal. Appl., 30:2 (1996), 106–117  mathnet  crossref  crossref  mathscinet  zmath  isi
    18. Lyons R., “Diffusions and Random Shadows in Negatively Curved Manifolds”, J. Funct. Anal., 138:2 (1996), 426–448  crossref  mathscinet  zmath  isi
    19. G. Carron, “Inégalités de hardy sur les variétés Riemanniennes non-compactes”, Journal de Mathématiques Pures et Appliquées, 76:10 (1997), 883  crossref
    20. Kuwae K., “Recurrence and Conservativeness of Symmetric Diffusion Processes by Girsanov Transformations”, J. Math. Kyoto Univ., 37:4 (1997), 735–756  mathscinet  zmath  isi
    21. Carron G., “Fraber-Krahn Inequalities and Sobolev-Orlicz Inclusion”, Potential Anal., 7:2 (1997), 555–575  crossref  mathscinet  zmath  isi
    22. A.G. Losev, “Elliptic partial differential equations on the warped products of Riemannian manifolds”, Applicable Analysis, 71:1-4 (1998), 325  crossref
    23. Chiung-Jue Sung, “Harmonic functions under quasi-isometry”, J Geom Anal, 8:1 (1998), 143  crossref
    24. Carron G., “An Exact Sequence in l-2-Cohomology”, Duke Math. J., 95:2 (1998), 343–372  crossref  mathscinet  zmath  isi
    25. Sung C., “A Note on the Existence of Positive Green's Function”, J. Funct. Anal., 156:1 (1998), 199–207  crossref  mathscinet  zmath  isi
    26. Lee Y., “Rough Isometry and Dirichlet Finite Harmonic Functions on Riemannian Manifolds”, Manuscr. Math., 99:3 (1999), 311–328  crossref  mathscinet  zmath  isi
    27. Losev A., Mazepa E., “Bounded Solutions of Shrodinger Equation on Noncompact Reimannian Manifolds of Special Type”, Dokl. Akad. Nauk, 367:2 (1999), 166–167  mathnet  mathscinet  zmath  isi
    28. Grigor'yan A., Saloff-Coste L., “Heat Kernel on Connected Sums of Riemannian Manifolds”, Math. Res. Lett., 6:3-4 (1999), 307–321  mathscinet  isi
    29. Grigor'yan A., “Analytic and Geometric Background of Recurrence and Non-Explosion of the Brownian Motion on Riemannian Manifolds”, Bull. Amer. Math. Soc., 36:2 (1999), 135–249  crossref  mathscinet  isi
    30. Holopainen I., “Volume Growth, Green's Functions, and Parabolicity of Ends”, Duke Math. J., 97:2 (1999), 319–346  crossref  mathscinet  zmath  adsnasa  isi
    31. Kim S., Lee Y., “Polynomial Growth Harmonic Functions on Connected Sums of Complete Riemannian Manifolds”, Math. Z., 233:1 (2000), 103–113  crossref  mathscinet  zmath  isi
    32. Gasymov M. Aslanov G., “Existence and the Asymptotic Behavior of Generalized Solutions of the Neumann Problem for Second-Order Elliptic Equations in Unbounded Layer Domains”, Differ. Equ., 37:12 (2001), 1699–1710  mathnet  crossref  mathscinet  zmath  isi
    33. Coulhon T., Holopainen L., Saloff-Coste L., “Harnack Inequality and Hyperbolicity for Subelliptic P-Laplacians with Applications to Picard Type Theorems”, Geom. Funct. Anal., 11:6 (2001), 1139–1191  crossref  mathscinet  zmath  isi
    34. Grigor'yan A., Kelbert M., “Asymptotic Separation for Independent Trajectories of Markov Processes”, Probab. Theory Relat. Field, 119:1 (2001), 31–69  crossref  mathscinet  isi
    35. Alexander Grigor'yan, Laurent Saloff-Coste, “Dirichlet heat kernel in the exterior of a compact set”, Comm Pure Appl Math, 55:1 (2002), 93  crossref  mathscinet  zmath
    36. Grigor'yan A., Saloff-Coste L., “Hitting Probabilities for Brownian Motion on Riemannian Manifolds”, J. Math. Pures Appl., 81:2 (2002), 115–142  crossref  mathscinet  isi
    37. Coulhon T., Grigor'yan A., Levin D., “On Isoperimetric Profiles of Product Spaces”, Commun. Anal. Geom., 11:1 (2003), 85–120  mathscinet  zmath  isi
    38. Takegoshi K., “A Note on Divergence of l-P-Integrals of Subharmonic Functions and its Applications”, Proc. Amer. Math. Soc., 131:9 (2003), 2849–2858  crossref  mathscinet  zmath  isi
    39. Lin Ch., Lu Zh., “On the Discrete Spectrum of Generalized Quantum Tubes”, Commun. Partial Differ. Equ., 31:10 (2006), 1529–1546  crossref  mathscinet  zmath  isi
    40. Grigor'yan A., “Heat Kernels on Weighted Manifolds and Applications”, Ubiquitous Heat Kernel, Contemporary Mathematics Series, 398, eds. Jorgenson J., Walling L., Amer Mathematical Soc, 2006, 93–191  crossref  mathscinet  isi
    41. Lin Ch., Lu Zh., “Existence of Bound States for Layers Built Over Hypersurfaces in Rn+1”, J. Funct. Anal., 244:1 (2007), 1–25  crossref  mathscinet  zmath  isi
    42. Yun G., “Stable Minimal Hypersurfaces in Locally Symmetric Spaces”, Math. Nachr., 280:15 (2007), 1744–1751  crossref  mathscinet  zmath  isi
    43. Pigola S., Rigoli M., Setti A.G., “Aspects of Potential Theory on Manifolds, Linear and Non-Linear”, Milan J. Math., 76:1 (2008), 229–256  crossref  mathscinet  zmath  isi
    44. Jie Xiao, “L[sup p]-Green potential estimates on noncompact Riemannian manifolds”, J Math Phys (N Y ), 51:6 (2010), 063515  crossref  elib
    45. Baer Ch., Bessa C.P., “Stochastic Completeness and Volume Growth”, Proc. Amer. Math. Soc., 138:7 (2010), 2629–2640  crossref  mathscinet  zmath  isi
    46. Howard S. Cohl, “Fundamental Solution of Laplace's Equation in Hyperspherical Geometry”, SIGMA, 7 (2011), 108, 14 pp.  mathnet  crossref
    47. Etesi G., Szabo S., “Harmonic Functions and Instanton Moduli Spaces on the Multi-Taub-NUT Space”, Commun. Math. Phys., 301:1 (2011), 175–214  crossref  mathscinet  zmath  adsnasa  isi
    49. Masamune J., Uemura T., Wang J., “On the Conservativeness and the Recurrence of Symmetric Jump-Diffusions”, J. Funct. Anal., 263:12 (2012), 3984–4008  crossref  mathscinet  zmath  isi
    50. Korolkov S.A., Losev A.G., “Generalized Harmonic Functions of Riemannian Manifolds with Ends”, Math. Z., 272:1-2 (2012), 459–472  crossref  mathscinet  zmath  isi
    51. Cohl H.S., Kalnins E.G., “Fourier and Gegenbauer Expansions for a Fundamental Solution of the Laplacian in the Hyperboloid Model of Hyperbolic Geometry”, J. Phys. A-Math. Theor., 45:14 (2012), 145206  crossref  mathscinet  zmath  adsnasa  isi  elib
    52. Cianchi A., Maz'ya V.G., “Bounds for Eigenfunctions of the Laplacian on Noncompact Riemannian Manifolds”, Am. J. Math., 135:3 (2013), 579–635  crossref  isi
    53. Kim J.-J., Yun G., “On the Structure of Complete Hypersurfaces in a Riemannian Manifold of Nonnegative Curvature and l (2) Harmonic Forms”, Arch. Math., 100:4 (2013), 369–380  crossref  mathscinet  zmath  isi
    54. Mao J., “A Class of Rotationally Symmetric Quantum Layers of Dimension 4”, J. Math. Anal. Appl., 397:2 (2013), 791–799  crossref  mathscinet  zmath  isi
    55. Lorenzo DʼAmbrosio, Serena Dipierro, “Hardy inequalities on Riemannian manifolds and applications”, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2013  crossref
    56. Jeong-Jin Kim, Gabjin Yun, “STRUCTURE OF STABLE MINIMAL HYPERSURFACES IN A Riemannian MANIFOLD OF NONNEGATIVE RICCI CURVATURE”, Bulletin of the Korean Mathematical Society, 50:4 (2013), 1201  crossref
    57. Grigor'yan A. Masamune J., “Parabolicity and Stochastic Completeness of Manifolds in Terms of the Green Formula”, J. Math. Pures Appl., 100:5 (2013), 607–632  crossref  isi
    58. N.T.hac Dung, Keomkyo Seo, “Vanishing theorems for <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="" xmlns:xs="" xmlns:xsi="" xmlns="" xmlns:ja="" xmlns:mml="" xmlns:tb="" xmlns:sb="" xmlns:ce="" xmlns:xlink="" xmlns:cals="" xmlns:sa=""><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> harmonic 1-forms on complete submanifolds in a Riemannian manifold”, Journal of Mathematical Analysis and Applications, 2014  crossref
    59. Yu. V. Goncharov, A. G. Losev, A. V. Svetlov, “Garmonicheskie funktsii na konusakh modelnykh mnogoobrazii”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2014, no. 3(22), 13–22  mathnet
    60. Yuhua Sun, “On the uniqueness of nonnegative solutions of differential inequalities with gradient terms on Riemannian manifolds”, CPAA, 14:5 (2015), 1743  crossref
    61. A. G. Losev, V. V. Filatov, “Teoremy tipa Liuvillya dlya reshenii statsionarnogo uravneniya Shredingera s konechnym integralom Dirikhle”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2016, no. 5(36), 13–23  mathnet  crossref
    62. A. A. Grigoryan, A. G. Losev, “O razmernosti prostranstv reshenii statsionarnogo uravneniya Shredingera na nekompaktnykh rimanovykh mnogoobraziyakh”, Matematicheskaya fizika i kompyuternoe modelirovanie, 20:3 (2017), 34–42  mathnet  crossref
    63. E. A. Mazepa, “O razreshimosti kraevykh zadach dlya uravneniya Puassona na nekompaktnykh rimanovykh mnogoobraziyakh”, Matematicheskaya fizika i kompyuternoe modelirovanie, 20:3 (2017), 136–147  mathnet  crossref
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
    Number of views:
    This page:526
    Full text:183

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019