On the Regularity of Weak Contact -Harmonic Maps

We prove Caccioppoli type estimates and consequently establish local H？lder continuity for a class of weak contact -harmonic maps from the Heisenberg group into the sphere . 1. Introduction The study of pseudoharmonic maps was started by Barletta et al. [1] (cf. also [2, 3] for successive investigations) as a generalization of the theory of harmonic maps among Riemannian manifolds (cf., e.g., [4]) and by identifying the results of Jost and Xu [5], Zhou [6], Haj？asz and Strzelecki [7], and Wang [8] as local aspects of the theory of pseudoharmonic maps from a strictly pseudoconvex CR manifold into a Riemannian manifold (cf. also [9, pages 225-226]). A similar class of maps, yet with values in another CR manifold, was studied in [10]. These are critical points of the functional where is a compact strictly pseudoconvex CR manifold of CR dimension ,？？ , and is a contact form on . Also is a contact Riemannian manifold and in particular an almost CR manifold (of CR codimension ). A moment's thought reveals the augmented difficulties such a theory may present. For instance, if and are two strictly pseudoconvex CR manifolds endowed, respectively, with contact forms and , then the pseudohermitian analog of the notion of a harmonic morphism (among Riemannian manifolds) is quite obvious: one may consider continuous maps such that the pullback of any local solution to in satisfies in in distribution sense. Here and are the sublaplacians of and , respectively. Unlike the situation in [2] (where the target manifold is Riemannian and pulls back local harmonic functions on to distribution solutions of ) such is not necessarily smooth (since it is unknown whether local coordinate systems on such that in might be produced). To give another example, should one look for a pseudohermitian analog to the Fluglede-Ishihara theorem (cf. [3] when is CR and is Riemannian), one would face the lack of an Ishihara type lemma (cf. [11]) as it is unknown whether admits local solutions whose (horizontal) gradient and hessian have prescribed values at a point. Moreover, what would be the appropriate notion of a hessian (cf. [12] for a possible choice)? A third example, discussed at some length in this paper, is that of the “degeneracy” of the Euler-Lagrange equations associated to the variational principle when is a Sasakian manifold. Indeed the matrix has but rank at each point (a well-known phenomenon in contact Riemannian geometry, cf., e.g., [13]. See also [14]). Consequently, in general one may not expect regularity of weak solutions to (2). For instance, if is the Heisenberg group