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.