A stochastic approach to a priori estimates and Liouville theorems for harmonic maps |
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Authors: | Anton Thalmaier Feng-Yu Wang |
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Affiliation: | 1. Unité de Recherche en Mathématiques, FSTC, Université du Luxembourg, 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg;2. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China;3. Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, UK |
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Abstract: | 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. |
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