Uniqueness of subelliptic harmonic maps
A map from a domain \(\Omega\subset\mathbb{R}^m\) to a Riemannian manifold \((N,g)\) is called a subelliptic harmonic map if it is a critical point of an energy functional \(E_X\) which is associated to a collection of vector fields \(X\). Here \(X=\{X_\alpha\}_{\alpha=1\ldots k}\) is assumed to satisfy the Hörmander condition, which is that the \(X_\alpha\) and all of their commutators up to a finite order span \(\mathbb{R}^m\). Then \(E_X(f):={1\over 2}\sum_\alpha\int_\Omega g\circ f(X_\alpha f, X_\alpha f) dx\). Since \(E_X\) is the energy in case \(X_\alpha=\partial_\alpha\), subelliptic harmonic maps generalize harmonic maps. In the current paper, the first and second variation of \(E_X\) are calculated (also in the case where \(\Omega\) is allowed to carry a Riemannian metric). Along the lines which are standard in harmonic map theory, this calculation leads to a partial uniqueness result if \(N\) has non-positive sectional curvature. By applying a subelliptic maximum principle, another uniqueness theorem for mappings to regular balls is proved. Two such subelliptic harmonic maps agree if they agree on the boundary of \(\Omega\), which must be non-characteristic with respect to \(X\) (a geometric condition which relates \(\partial\Omega\) and \(X\)).
scientific article; zbMATH DE number 1396332