A remark on the maximum principle and stochastic completeness
Authors:
Stefano Pigola, Marco Rigoli and Alberto G. Setti
Journal:
Proc. Amer. Math. Soc. 131 (2003), 12831288
MSC (2000):
Primary 58J35; Secondary 58J65
Published electronically:
July 26, 2002
MathSciNet review:
1948121
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Abstract: We prove that the stochastic completeness of a Riemannian manifold is equivalent to the validity of a weak form of the OmoriYau maximum principle. Some geometric applications of this result are also presented.
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 [A]
 R. Azencott, Behaviour of diffusion semigroups at infinity, Bull. Soc. Math. France 102 (1974), 193240. MR 50:8725
 [CR]
 P. Collin and H. Rosenberg, Notes sur la démonstration de Nadirashvili des conjectures de Hadamard et CalabiYau, Bull. Sc. Math. 123 (1999), 563575. MR 2000i:53011
 [CX]
 Q. Chen and Y.L. Xin, A generalized maximum principle and its applications in geometry, Amer. J. Math. 114 (1992), 355366. MR 93g:53054
 [CY]
 S.Y. Cheng and S.T. Yau, Differential equations on Riemanninan manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), 333354. MR 52:6608
 [EK]
 J. Eells and S. Kobayashi, Problems in differential geometry. In Proc. of USJapan Seminar on differential geometry, Kyoto (1965), 167177.
 [EL]
 J. Eells and L. Lemaire, Selected Topics in Harmonic Maps, CBMS n. 50, AMS, Providence, 1983. MR 85g:58030
 [FC]
 D. FischerColbrie, Some rigidity theorems for minimal submanifolds of the sphere, Acta Math. 145 (1980), 2946. MR 82b:53078
 [G]
 M.P. Gaffney, The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math. 12 (1959), 111. MR 21:892
 [Gr1]
 A. Grigor'yan, On stochastically complete manifolds, Engl. Transl. Soviet Math. Dokl. 34 (1987), 310313. MR 88a:58209
 [Gr2]
 A. Grigor'yan, Bounded solutions of the Schrödinger equation on noncompact Riemannian manifolds, Engl. Transl. J. of Soviet Math. 51 (1990), 23402349. MR 90m:35050
 [Gr3]
 A. Grigor'yan, Analytic and geometric background of recurrence and nonexplosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), 135249. MR 99k:58195
 [GW]
 R. Greene and H.H. Wu, Function Theory on Manifolds Which Possess a Pole, Lecture Notes in Math. n. 699, Springer Verlag, Berlin, 1979 MR 81a:53002
 [HM]
 D. Hoffman and W. Meeks, The strong halfspace theorem for minimal surfaces, Invent. Math. 101 (1990), 373377. MR 92e:53010
 [K]
 L. Karp, Differential inequalities on complete Riemannian manifolds and applications, Math. Ann. 272 (1985), 449459. MR 87g:58119
 [Ks]
 A. Kasue, Estimates for solutions of Poisson equations and their applications to submanifolds, Springer Lecture Notes in Mathematics 1090 (1984), 114. MR 86d:58118
 [L]
 P. Li, Uniqueness of solutions for the Laplace equation and the heat equation on Riemannian manifolds, J. Diff. Geom. 20 (1984), 447457. MR 86h:58133
 [MY]
 N. Mok and S.T. Yau, Completeness of the KählerEinstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions, Proc. Symp. Pure Math. 39 (1983), 4159. MR 85j:53068
 [N]
 N.S. Nadirashvili, Hadamard's and CalabiYau's conjectures on negatively curved and minimal surfaces, Invent. Math. 126 (1996), 457465. MR 98d:53014
 [O]
 H. Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 (1967), 205214. MR 35:6101
 [R]
 H.L. Royden, The AhlforsSchwarz lemma in several complex variables, Comm. Math. Helv. 55 (1980), 547558. MR 82i:32049
 [RRS]
 A. Ratto, M. Rigoli and A.G. Setti, On the OmoriYau maximum principle and its applications to differential equations and geometry, J. Funct. Anal. 134 (1995), 486510. MR 96k:53062
 [RRV]
 A. Ratto, M. Rigoli and L. Veron, Scalar curvature and conformal deformation of hyperbolic space, J. Funct. Anal. 121 (1994), 1577. MR 95a:53062
 [T]
 K. Takegoshi, A volume estimate for strong subharmonicity and maximum principle on complete Riemannian manifolds, Nagoya Math. J. 151 (1998), 2536. MR 99i:53042
 [Y1]
 S.T. Yau, On the heat kernel of a complete Riemannian manifold, J. Math. Pures Appl. 57 (1978), 191201. MR 81b:58041
 [Y2]
 S.T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201228. MR 55:4042
 [Y3]
 S.T. Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978), 197203. MR 58:6370
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Additional Information
Stefano Pigola
Affiliation:
Dipartimento di Matematica, Università di Milano, via Saldini 50, I20133 Milano, Italy
Email:
pigola@mat.unimi.it
Marco Rigoli
Affiliation:
Dipartimento di Scienze C.F.M., Università dell’Insubria  Como, via Valleggio 11, I22100 Como, Italy
Email:
rigoli@matapp.unimib.it
Alberto G. Setti
Affiliation:
Dipartimento di Scienze C.F.M., Università dell’Insubria  Como, via Valleggio 11. I22100 Como, Italy
Email:
setti@uninsubria.it
DOI:
http://dx.doi.org/10.1090/S0002993902066728
PII:
S 00029939(02)066728
Keywords:
Maximum principle,
stochastic completeness
Received by editor(s):
January 3, 2001
Received by editor(s) in revised form:
November 1, 2001
Published electronically:
July 26, 2002
Dedicated:
Dedicated to the memory of Franca Burrone Rigoli
Communicated by:
Bennett Chow
Article copyright:
© Copyright 2002
American Mathematical Society
