On subelliptic harmonic maps with potential

TL;DR

This paper studies the existence of subelliptic harmonic maps with potential on sub-Riemannian manifolds using a heat flow approach, extending classical results to more general geometric settings.

Contribution

It introduces a new existence theory for subelliptic harmonic maps with potential on step-$2$ and step-$r$ sub-Riemannian manifolds with non-positive curvature targets.

Findings

Proves existence results under non-positive curvature assumptions

Establishes conditions on the potential function G

Extends classical harmonic map theory to subelliptic setting

Abstract

Let $(M,H,g_H;g)$ be a sub-Riemannian manifold and $(N,h)$ be a Riemannian manifold. For a smooth map $u: M \to N$, we consider the energy functional $E_G(u) = \frac{1}{2} \int_M[|\mathrm{d}u_H|^2-2G(u)] \mathrm{d}V_M$, where $\mathrm{d}u_H$ is the horizontal differential of $u$, $G:N\to \mathbb{R}$ is a smooth function on $N$. The critical maps of $E_G(u)$ 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 non-positive 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.