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**1 - 4**of**4**### ScienceDirect A stochastic approach to the harmonic map heat flow on manifolds with time-dependent Riemannian metric

"... 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 glo ..."

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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.

### 4 A NOTE ON LOWER DIAMETER BOUNDS FOR CLOSED DOMAIN MANIFOLDS OF SHRINKING RICCI-HARMONIC SOLITONS

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