X-Elliptic Harmonic Maps

Abstract

We study X-elliptic harmonic maps of an open set $\mathcal{U}\subset \mathbb{R}^{N}$ endowed with a family of vector fields X = { X ₁, ⋯ , 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 Gutiérrez and Lanconelli (Commun Partial Differ Equ 28:1833–1862, 2003) we prove openness of X-elliptic harmonic morphisms.

Dedicated to Ermanno Lanconelli on the occasion of his 70th birthday

AMS Classification: Primary: 53C43, Secondary: 35H20