A probabilistic proof of S.-Y. Cheng’s Liouville theorem. (English) Zbl 0718.58015

From the author’s abstract: “Let f: \(M\to N\) 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 proved analytically by S. Y. Cheng. This paper describes a probabilistic proof under the same hypotheses.”


58E20 Harmonic maps, etc.
58J65 Diffusion processes and stochastic analysis on manifolds
60J65 Brownian motion
53C20 Global Riemannian geometry, including pinching
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