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We announce several results of an extensive study [A] of the size of singular sets in oriented m dimensional surfaces which are area minimizing in m + I dimensional Riemannian manifolds. Our principal result is that the Hausdorff dimension of such a singular set does not exceed ra — 2. Examples show this is the best possible such general estimate when / > 2, i.e., when branching singularities «are possible. The general existence of such surfaces of least area is well known in a variety of settings [F, 5.1.6]. In order to obtain estimates on branching of area minimizing surfaces we were led to use Taylor's expansion in terms of first derivatives at 0 to approximate the nonparametric area integrand by Dirichlet's integrand. Accordingly, we study branched coverings of regions in R m which are graphs of multiple valued functions minimizing the integral of Dirichlet's integrand. As a central estimate in our analysis of area minimizing surfaces we show that the Hausdorff dimension of the branch set of such a minimizing covering does not exceed m — 2. To state several results in more detail we use the terminology of [F]. Suppose that A is a bounded open subset of R m with smooth boundary, and let k, /, ra, n, Q be positive integers with k > 3, I < n, and m > 2.

𝑄 valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two