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.
Citation
Seth Stafford. "A Probabilistic Proof of S.-Y. Cheng's Liouville Theorem." Ann. Probab. 18 (4) 1816 - 1822, October, 1990. https://doi.org/10.1214/aop/1176990651
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