A stochastic approach to a priori estimates and Liouville theorems for harmonic maps

作者：Thalmaier, Anton^*; Wang, Feng-Yu

来源：Bulletin des Sciences Mathematiques, 2011, 135(6-7): 816-843.

DOI：10.1016/j.bulsci.2011.07.014

• Liouville theorem
• Harmonic map
• Gradient estimate
• Ricci curvature
• Stochastic development
• Brownian motion
• Nonlinear martingale
• Bismut formula

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.

• 出版日期2011-11
• 单位北京师范大学