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

## TL;DR

This paper extends Eells-Sampson theorems to subelliptic harmonic maps from sub-Riemannian manifolds, proving existence results for the associated heat flow under curvature conditions.

## Contribution

It introduces existence results for subelliptic harmonic maps from sub-Riemannian manifolds using heat flow methods, generalizing classical theorems to a subelliptic setting.

## Key findings

- Existence of subelliptic harmonic maps under non-positive curvature.
- Eells-Sampson type theorems for step-2 and step-r sub-Riemannian manifolds.
- Hartman type results for the subelliptic harmonic map flow.

## Abstract

In this paper, we consider critical maps of a horizontal energy functional for maps from a sub-Riemannian manifold to a Riemannian manifold. These critical maps are referred to as subelliptic harmonic maps. In terms of the subelliptic harmonic map heat flow, we investigate the existence problem for subelliptic harmonic maps. Under the assumption that the target Riemannian manifold has non-positive sectional curvature, we prove some Eells-Sampson type existence results for this flow 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. Finally, some Hartman type results are also established for the flow.