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), 1283-1288

MSC (2000):
Primary 58J35; Secondary 58J65

Published electronically:
July 26, 2002

MathSciNet review:
1948121

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the stochastic completeness of a Riemannian manifold is equivalent to the validity of a weak form of the Omori-Yau maximum principle. Some geometric applications of this result are also presented.

**[A]**Robert Azencott,*Behavior of diffusion semi-groups at infinity*, Bull. Soc. Math. France**102**(1974), 193–240. MR**0356254****[CR]**Pascal Collin and Harold Rosenberg,*Notes sur la démonstration de N. Nadirashvili des conjectures de Hadamard et Calabi-Yau*, Bull. Sci. Math.**123**(1999), no. 7, 563–575 (French). MR**1713304**, 10.1016/S0007-4497(99)00113-X**[CX]**Q. Chen and Y. L. Xin,*A generalized maximum principle and its applications in geometry*, Amer. J. Math.**114**(1992), no. 2, 355–366. MR**1156569**, 10.2307/2374707**[CY]**S. Y. Cheng and S. T. Yau,*Differential equations on Riemannian manifolds and their geometric applications*, Comm. Pure Appl. Math.**28**(1975), no. 3, 333–354. MR**0385749****[EK]**J. Eells and S. Kobayashi,*Problems in differential geometry.*In Proc. of US-Japan Seminar on differential geometry, Kyoto (1965), 167-177.**[EL]**James Eells and Luc Lemaire,*Selected topics in harmonic maps*, CBMS Regional Conference Series in Mathematics, vol. 50, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983. MR**703510****[FC]**D. Fischer-Colbrie,*Some rigidity theorems for minimal submanifolds of the sphere*, Acta Math.**145**(1980), no. 1-2, 29–46. MR**558091**, 10.1007/BF02414184**[G]**Matthew P. Gaffney,*The conservation property of the heat equation on Riemannian manifolds.*, Comm. Pure Appl. Math.**12**(1959), 1–11. MR**0102097****[Gr1]**A. A. Grigor′yan,*Stochastically complete manifolds*, Dokl. Akad. Nauk SSSR**290**(1986), no. 3, 534–537 (Russian). MR**860324****[Gr2]**A. A. Grigor′yan,*Bounded solutions of the Schrödinger equation on noncompact Riemannian manifolds*, Trudy Sem. Petrovsk.**14**(1989), 66–77, 265–266 (Russian, with English summary); English transl., J. Soviet Math.**51**(1990), no. 3, 2340–2349. MR**1001354**, 10.1007/BF01094993**[Gr3]**Alexander Grigor′yan,*Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds*, Bull. Amer. Math. Soc. (N.S.)**36**(1999), no. 2, 135–249. MR**1659871**, 10.1090/S0273-0979-99-00776-4**[GW]**R. E. Greene and H. Wu,*Function theory on manifolds which possess a pole*, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. MR**521983****[HM]**D. Hoffman and W. H. Meeks III,*The strong halfspace theorem for minimal surfaces*, Invent. Math.**101**(1990), no. 2, 373–377. MR**1062966**, 10.1007/BF01231506**[K]**Leon Karp,*Differential inequalities on complete Riemannian manifolds and applications*, Math. Ann.**272**(1985), no. 4, 449–459. MR**807283**, 10.1007/BF01455859**[Ks]**Atsushi Kasue,*Estimates for solutions of Poisson equations and their applications to submanifolds*, Differential geometry of submanifolds (Kyoto, 1984) Lecture Notes in Math., vol. 1090, Springer, Berlin, 1984, pp. 1–14. MR**775140**, 10.1007/BFb0101562**[L]**Peter Li,*Uniqueness of 𝐿¹ solutions for the Laplace equation and the heat equation on Riemannian manifolds*, J. Differential Geom.**20**(1984), no. 2, 447–457. MR**788288****[MY]**Ngaiming Mok and Shing-Tung Yau,*Completeness of the Kähler-Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions*, The mathematical heritage of Henri Poincaré, Part 1 (Bloomington, Ind., 1980) Proc. Sympos. Pure Math., vol. 39, Amer. Math. Soc., Providence, RI, 1983, pp. 41–59. MR**720056****[N]**Nikolai Nadirashvili,*Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimal surfaces*, Invent. Math.**126**(1996), no. 3, 457–465. MR**1419004**, 10.1007/s002220050106**[O]**Hideki Omori,*Isometric immersions of Riemannian manifolds*, J. Math. Soc. Japan**19**(1967), 205–214. MR**0215259****[R]**H. L. Royden,*The Ahlfors-Schwarz lemma in several complex variables*, Comment. Math. Helv.**55**(1980), no. 4, 547–558. MR**604712**, 10.1007/BF02566705**[RRS]**Andrea Ratto, Marco Rigoli, and Alberto G. Setti,*On the Omori-Yau maximum principle and its applications to differential equations and geometry*, J. Funct. Anal.**134**(1995), no. 2, 486–510. MR**1363809**, 10.1006/jfan.1995.1154**[RRV]**Andrea Ratto, Marco Rigoli, and Laurent Véron,*Scalar curvature and conformal deformation of hyperbolic space*, J. Funct. Anal.**121**(1994), no. 1, 15–77. MR**1270588**, 10.1006/jfan.1994.1044**[T]**Kensho Takegoshi,*A volume estimate for strong subharmonicity and maximum principle on complete Riemannian manifolds*, Nagoya Math. J.**151**(1998), 25–36. MR**1650344****[Y1]**Shing Tung Yau,*On the heat kernel of a complete Riemannian manifold*, J. Math. Pures Appl. (9)**57**(1978), no. 2, 191–201. MR**505904****[Y2]**Shing Tung Yau,*Harmonic functions on complete Riemannian manifolds*, Comm. Pure Appl. Math.**28**(1975), 201–228. MR**0431040****[Y3]**Shing Tung Yau,*A general Schwarz lemma for Kähler manifolds*, Amer. J. Math.**100**(1978), no. 1, 197–203. MR**0486659**

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Additional Information

**Stefano Pigola**

Affiliation:
Dipartimento di Matematica, Università di Milano, via Saldini 50, I-20133 Milano, Italy

Email:
pigola@mat.unimi.it

**Marco Rigoli**

Affiliation:
Dipartimento di Scienze C.F.M., Università dell’Insubria - Como, via Valleggio 11, I-22100 Como, Italy

Email:
rigoli@matapp.unimib.it

**Alberto G. Setti**

Affiliation:
Dipartimento di Scienze C.F.M., Università dell’Insubria - Como, via Valleggio 11. I-22100 Como, Italy

Email:
setti@uninsubria.it

DOI:
https://doi.org/10.1090/S0002-9939-02-06672-8

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