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

Stafford, Seth

Ann. Probab. 18, No. 4, 1816-1822 (1990).

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.”

Reviewer: Pan Yanglian (Shanghai)

MSC:

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

Keywords:

Brownian motion; harmonic map; Riemannian manifolds; Ricci curvature