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In this paper, we consider critical maps of a horizontal energy functionalfor maps from a sub-Riemannian manifold to a Riemannian manifold. Thesecritical maps are referred to as subelliptic harmonic maps. In terms of thesubelliptic harmonic map heat flow, we investigate the existence problem forsubelliptic harmonic maps. Under the assumption that the target Riemannianmanifold has non-positive sectional curvature, we prove some Eells-Sampson typeexistence results for this flow when the source manifold is either a step-2sub-Riemannian manifold or a step-r sub-Riemannian manifold whosesub-Riemannian structure comes from a tense Riemannian foliation. Finally, someHartman type results are also established for the flow.

Eells-Sampson type theorems for subelliptic harmonic maps from sub-Riemannian manifolds