An important question in regularity theory of minimal submanifolds is that of understanding the nature of weakly defined minimal submanifolds (stationary integral currents or varifolds) in the presence of branch points, namely, those singular points where there are planar tangent cones of multiplicity > 1. Related is the question of what natural geometric conditions would prevent the existence of such points. In regions where branch points can be ruled out, one can conclude that the singular set in general has codimension at least 1, and higher under various additional hypotheses. When branch points exist, it is of interest to know how large a set they form and what the local structure of the varifold near them is.

These lectures will focus on a new regularity theorem for the class of stationary, stable codimension 1 integral varifolds of an open ball. Making no assumption a priori on the size of the singular set, this theorem gives a rather natural, "checkable" geometric structural condition for non-existence of branch points in varifolds in this class. Specifically, this condition requires that no singular point has a neighborhood in which the varifold corresponds to a union of (three or more) C^1,α (for some arbitrarily chosen α ∈ (0, 1)) hypersurfaces-with-boundary meeting (only) along their common boundary. The conclusion when this condition is satisfied is that the stable varifold corresponds to a smooth embedded hypersurface away from the boundary of the ball and away from a possible interior singular set of codimension at least 7 (which is empty if the dimension of the varifold is ≤ 6). The work generalizes the regularity theory of R. Schoen and L. Simon.

In the absence of this structural condition, branch points, in general, do exist in stable hypersurfaces. A theorem giving a fairly complete description of the local structure of a codimension 1 stable integral current, including an optimal estimate on the size of its branch set, is also available provided the multiplicity of the current is at most 2. For the bound on the size of the branch set, the latter theorem relies on recent joint work with Leon Simon on estimating the Hausdorff dimension of the set of order 2 branch points of a C^1,α branched minimal graph (of arbitrary codimension). These results will also be discussed.

Lectures 2 and 3 will partly be devoted to describing some of the techniques involved in the proofs of the main theorems; Lecture 1 will be less so and will contain general introductory material.