Self-intersection points in classical area-minimizing surfaces (Q1895743)

The author studies the structure at a point of tangential self- intersection in an area-minimizing classical minimal surface \(M\) of genus \(g\) in \(\mathbb{R}^n\). Here a point of tangential self-intersection is a point of self-intersection of \(M\) where two pieces of the surface meet tangentially, and in a neighbourhood of this point \(M\) consists of two minimzing discs, the union of which is also minimizing among oriented surfaces with a (possibly degenerate) handle. Let \(F : D_1 \cup D_2 \subset \mathbb{C} \to \mathbb{R}^n\) be an area-minimizing immersion from two discs with a point of tangency. Without loss of generality, the \(D_i\) have non-parametric representations: \(z \mapsto (z, f_i (z)) \in \mathbb{C} \times \mathbb{R}^{n - 2}\), \(i = 1,2\), for \(z\) near zero, where \(f_i : D \to \mathbb{R}^{n - 2}\) are real analytic and \(f_i (z) = 0(|z|)\). The main result of this paper is that the lowest order homogeneous term \(p(z)\) of the splitting function \(h = f_2 - f_1\) has the form \(p(z) = az^k + \overline {a} \overline {z}^k\), where \(p : D \subset \mathbb{C} \to \mathbb{R}^{n - 2}\), \(0 \in D\) is the point of tangency and \(a \in \mathbb{C}^{n - 2}\) with \(a \cdot a = 0\). Moreover, the author also extends the result to a point of tangential self-intersection which may be a branch point on one or more of the discs.