The Annals of Probability

A Probabilistic Proof of S.-Y. Cheng's Liouville Theorem

Seth Stafford

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Abstract

Let $f: M \rightarrow N$ be a harmonic map between complete Riemannian manifolds $M$ and $N$, and suppose the Ricci curvature of $M$ is nonnegative definite, the sectional curvature of $N$ is nonpositive, and $N$ is simply connected. Then if $f$ has sublinear asymptotic growth, $f$ must be a constant map. This result was first proved analytically by S.-Y. Cheng. This paper describes a probabilistic proof under the same hypotheses.

Article information

Source
Ann. Probab. Volume 18, Number 4 (1990), 1816-1822.

Dates
First available: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176990651

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aop/1176990651

Mathematical Reviews number (MathSciNet)
MR1071828

Zentralblatt MATH identifier
0718.58015

Subjects
Primary: 58G32
Secondary: 60J65: Brownian motion [See also 58J65]

Keywords
Riemannian manifolds Brownian motion Ricci curvature harmonic maps

Citation

Stafford, Seth. A Probabilistic Proof of S.-Y. Cheng's Liouville Theorem. The Annals of Probability 18 (1990), no. 4, 1816--1822. doi:10.1214/aop/1176990651. http://projecteuclid.org/euclid.aop/1176990651.


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