## The Annals of Probability

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

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

#### 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 in Project Euclid: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176990651

**Digital Object Identifier**

doi:10.1214/aop/1176990651

**Mathematical Reviews number (MathSciNet)**

MR1071828

**Zentralblatt MATH identifier**

0718.58015

**JSTOR**

links.jstor.org

**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. Ann. Probab. 18 (1990), no. 4, 1816--1822. doi:10.1214/aop/1176990651. http://projecteuclid.org/euclid.aop/1176990651.