Legendrian energy minimizers. Part I: Heisenberg group target

摘要

In their joint work [18] Korevaar and Schoen introduced a definition of Sobolev spaces of functions with target in a metric space. This definition is reduced to the traditional one [23] when the metric space is a Riemannian manifold equipped with the geodesic distance, but it has the advantage of being independent from the choice of the isometric embedding of the manifold into an Euclidean space. Another feature of the Korevaar and Schoen's definition is that it allows for a satisfactory existence and regularity theory for energy minimizers whenever the target space has non-positive curvature (in the sense of Alexandrov). See also [14] and [19] for related results. In general, in absence of conditions on the curvature, one does not have neither existence nor regularity of the minimizers.