Tohoku Mathematical Journal

Stokes' theorem, volume growth and parabolicity

Daniele Valtorta and Giona Veronelli

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We present some new Stokes' type theorems on complete non-compact manifolds that extend, in different directions, previous works by Gaffney and Karp and also the so called Kelvin-Nevanlinna-Royden criterion for $p$-parabolicity. Applications to comparison and uniqueness results involving the $p$-Laplacian are deduced.

Article information

Source
Tohoku Math. J. (2) Volume 63, Number 3 (2011), 397-412.

Dates
First available: 11 October 2011

Permanent link to this document
http://projecteuclid.org/euclid.tmj/1318338948

Digital Object Identifier
doi:10.2748/tmj/1318338948

Mathematical Reviews number (MathSciNet)
MR2851103

Subjects
Primary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc.
Secondary: 31C45: Other generalizations (nonlinear potential theory, etc.)

Keywords
$p$-parabolicity Stokes' theorem Kelvin-Nevanlinna-Royden criterion

Citation

Valtorta, Daniele; Veronelli, Giona. Stokes' theorem, volume growth and parabolicity. Tohoku Mathematical Journal 63 (2011), no. 3, 397--412. doi:10.2748/tmj/1318338948. http://projecteuclid.org/euclid.tmj/1318338948.


Export citation

References

  • H.-D. Cao, Y. Shen and S. Zhu, The structure of stable minimal hypersurfaces in $\boldsymbolR^n+1$, Math. Res. Lett. 4 (1997), 637–644.
  • G. Carron, Inégalités isopérimétriques et inégalités de Faber-Krahn. Séminaire de Théorie Spectrale et Géométrie, No. 13, Année 1994–1995, 63–66.
  • I. Chavel, Riemannian geometry: A modern introduction, Cambridge Tracts in Math. 108, Cambridge University Press, Cambridge, 1993.
  • M. Gaffney, A special Stokes' Theorem for complete Riemannian manifolds, Ann. of Math. 60 (1954), 140–145.
  • A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999),135–249.
  • A. Grigor'yan, Isoperimetric inequalities and capacities on Riemannian manifolds The Mazya anniversary collection, Vol. 1 (Rostock, 1998), 139–153, Oper. Theory Adv. Appl, 109, Birkhäuser, Basel, 1999.
  • V. Gol'dshtein and M. Troyanov, The Kelvin-Nevanlinna-Royden criterion for $p$-parabolicity, Math Z. 232 (1999), 607–619.
  • R. Greene and H. H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Math. 699, Springer Verlag, Berlin, 1979.
  • R. Hardt and F.-H. Lin, Mappings minimizing the $L^p$ norm of the gradient, Comm. Pure Appl. Math. 40 (1987), 555–588.
  • I. Holopainen, Nonlinear potential theory and quasiregular mappings on Riemannian manifolds, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 74 (1990), 45 pp.
  • I. Holopainen, Quasiregular mappings and the $p$-Laplace operator, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), 219–239, Contemp. Math. 338, Amer. Math. Soc., Providence, RI, 2003.
  • I. Holopainen, S. Pigola and G. Veronelli, Global comparison principles for the $p$-Laplace operator on Riemannian manifolds, Potential Analysis 34 (2011), 371–384.
  • L. Karp, On Stokes' Theorem for noncompact manifolds, Proc. Amer. Math. Soc. 82 (1981), 487–490.
  • P. Li and J. Wang, Minimal hypersurfaces with finite index, Math. Res. Lett. 9 (2002), 95–103.
  • P. Lindqvist, On the equation $\operatornamediv (\vert \nabla u \vert ^p-2\nabla u) +\lambda \vert u \vert ^p-2u=0$, Proc. Amer Math. Soc. 109 (1990), 157–164.
  • T. Lyons and D. Sullivan, Function theory, random paths and covering spaces, J. Differential Geom. 19 (1984), 229–323.
  • S. Pigola, M. Rigoli and A. G. Setti, Constancy of $p$-harmonic maps of finite $q$-energy into non-positively curved manifolds, Math. Z. 258 (2008), 347–362.
  • S. Pigola, M. Rigoli and A. G. Setti, Vanishing and finiteness results in geometric analysis, A generalization of the Bochner technique, Prog. Math. 266 (2008), Birkhäuser Verlag, Basel 2008.
  • S. Pigola, A. G. Setti and M. Troyanov, The topology at infinity of a manifold supporting an $L^p,q$-Sobolev inequality, Submitted. Preliminary version: arXiv:1007.1761v1
  • M. Rigoli and A. G. Setti, Liouville type theorems for $\varphi$-subharmonic functions, Rev. Mat. Iberoamericana 17 (2001), 471–520.
  • L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Bomd 164, Springer Verlag, New York-Berlin, 1970.
  • R. Schoen and S. T. Yau, Compact group actions and the topology of manifolds with nonpositive curvature, Topology 18 (1979), 361–380.
  • R. Schoen and S. T. Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature, Comment. Math. Helv. 51 (1976), 333–341.
  • P. Tolksdorf, Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pura Appl. (4) 134 (1983), 241–266.
  • M. Troyanov, Parabolicity of manifolds, Siberian Adv. Math. 9 (1999), 125–150.
  • D. Valtorta, Potenziali di Evans su varietà paraboliche, Master Thesis, 2009.
  • S. W. Wei and C.-M. Yau, Regularity of $p$-energy minimizing maps and $p$-superstrongly unstable indices, J. Geom. Anal. 4 (1994), 247–272.