On subelliptic harmonic maps with potential

Let \((M,H,g_H;g)\) $(M,H,g_H;g)$ be a sub-Riemannian manifold and (N, h) be a Riemannian manifold. For a smooth map \(u: M \rightarrow N\) $u:M\to N$ , we consider the energy functional \(E_G(u) = \frac{1}{2} \int _M[|\textrm{d}u_\text {H}|^2 - 2\,G(u)] \textrm{d}V_M\) $E_G(u)=\frac{1}{2}{\int }_M[|\text{d}{u}_{\text{H}}{|}^2-2G(u)]\text{d}{V}_M$ , where \(\textrm{d}u_\text {H}\) $\text{d}{u}_{\text{H}}$ is the horizontal differential of u, \(G:N\rightarrow \mathbb {R}\) $G:N\to \mathbb{R}$ is a smooth function on N. The critical maps of \(E_G(u)\) $E_G(u)$ are referred to as subelliptic harmonic maps with potential G. In this paper, we investigate the existence problem for subelliptic harmonic maps with potentials by a subelliptic heat flow. Assuming that the target Riemannian manifold has nonpositive sectional curvature and the potential G satisfies various suitable conditions, we prove some Eells–Sampson-type existence results when the source manifold is either a step-2 sub-Riemannian manifold or a step-r sub-Riemannian manifold whose sub-Riemannian structure comes from a tense Riemannian foliation.