A Probabilistic Proof of S.-Y. Cheng's Liouville Theorem
Stafford, Seth
Ann. Probab., Tome 18 (1990) no. 4, p. 1816-1822 / Harvested from Project Euclid
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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.
Publié le : 1990-10-14
Classification:  Riemannian manifolds,  Brownian motion,  Ricci curvature,  harmonic maps,  58G32,  60J65 
@article{1176990651,
     author = {Stafford, Seth},
     title = {A Probabilistic Proof of S.-Y. Cheng's Liouville Theorem},
     journal = {Ann. Probab.},
     volume = {18},
     number = {4},
     year = {1990},
     pages = { 1816-1822},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176990651}
}

Stafford, Seth. A Probabilistic Proof of S.-Y. Cheng's Liouville Theorem. Ann. Probab., Tome 18 (1990) no. 4, pp.  1816-1822. http://gdmltest.u-ga.fr/item/1176990651/