X-Elliptic Harmonic Maps

We study X-elliptic harmonic maps of an open set \(\mathcal{U}\subset \mathbb{R}^{N}\) endowed with a family of vector fields X = { X1, ⋯ , X m } into a Riemannian manifold S i.e. \(C^{\infty }\) solutions \(\phi: \mathcal{U}\rightarrow S\) to the nonlinear system \(-L\phi ^{\alpha } + a^{ij}(\varGamma _{\beta \gamma }^{\alpha }\circ \phi )(\partial _{x^{i}}\phi ^{\beta })(\partial _{ x^{j}}\phi ^{\gamma }) = 0\) where \(L =\sum _{ i,j=1}^{N}\partial _{x_{j}}(a^{ij}(x)\,\partial _{x_{j}}u)\) is an uniformly X-elliptic operator. We establish a Solomon type (cf. Solomon, J Differ Geom 21:151–162, 1985) result for X-elliptic harmonic maps \(\phi: \mathcal{U}\rightarrow S^{M}\setminus \varSigma\) with values into a sphere and omitting a codimension two totally geodesic submanifold \(\varSigma \subset S^{M}\). As an application of Harnack inequality (for positive solutions to Lu = 0) in Gutierrez and Lanconelli (Commun Partial Differ Equ 28:1833–1862, 2003) we prove openness of X-elliptic harmonic morphisms.