A Stochastic Approach to A Priori Estimates and Liouville Theorems
for Harmonic Maps
Nonlinear versions of Bismut type formulas for the differential of a
harmonic map between Riemannian manifolds are used to establish a priori
estimates for harmonic maps.
A variety of Liouville type theorems is shown to follow as corollaries
from such estimates by exhausting the domain through an increasing
sequence of geodesic balls.
This probabilistic method is well suited for proving sharp estimates under
various curvature conditions.
We discuss Liouville theorems for harmonic maps under the following conditions:
small image, sublinear growth, non-positively curved targets, generalized
bounded dilatation, Liouville manifolds as domains, certain asymptotic behaviour.
by Anton Thalmaier and Feng-Yu Wang
Bull. Sci. Math. 135 (2011) 816-843
The paper is available here:
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