A stochastic approach to the harmonic map heat flow on manifolds with time-dependent Riemannian metric |
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| Authors: | Hongxin Guo Robert Philipowski Anton Thalmaier |
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| Affiliation: | 1. School of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, China;2. Unité de Recherche en Mathématiques, Université du Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359 Luxembourg, Luxembourg |
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| Abstract: | We first prove stochastic representation formulae for space–time harmonic mappings defined on manifolds with evolving Riemannian metric. We then apply these formulae to derive Liouville type theorems under appropriate curvature conditions. Space–time harmonic mappings which are defined globally in time correspond to ancient solutions to the harmonic map heat flow. As corollaries, we establish triviality of such ancient solutions in a variety of different situations. |
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| Keywords: | 53C44 58J65 |
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