A theory of multiholomorphic maps

Abstract

This paper presents and explores a theory of \emph{multiholomorphic maps}. This group of ideas generalizes the theory of pseudoholomorphic curves in a direction suggested by consideration of the kinds of compatible geometric structures that appear in the realm of special holonomy as well as some of the topological and analytic considerations that are essential to pseudoholomorphic invariants. The first part presents the geometric framework of compatible $n$-triads, from which follows naturally the definition of a multiholomorphic mapping. Some of the general analytic and differential-geometric properties of these maps are derived, including an energy identity which expresses a multiholomorphic map as a minimizer in its homotopy class of the appropriate $L^p$-energy. Some theorems confining the critical loci of such maps are obtained as well as some Liouville-type theorems for maps with sufficient regularity in the presence of curvature hypotheses. Finally, attention is focused onto a special case of the theory which pertains to the calibrated geometry of $\Gtwo$-manifolds.