Branching point (of a minimal surface)

A singular point of a minimal surface in which the first fundamental form of the surface vanishes; this means, in fact, that such a branching point can exist on a generalized minimal surface only. This singular point owes its name to the fact that in a neighbourhood of it the structure of the generalized minimal surface resembles that of the Riemann surface of the function $w=z^n$, $n\geq2$, over the point $z=0$, i.e. there the generalized minimal surface has a many-sheeted orthogonal projection onto some plane domain, in which the projection of the branching point itself is an interior point with a unique pre-image. In a neighbourhood of a branching point $(u=0,v=0)$ the coordinates $(x,y,z)$ of the minimal surface can be represented in the form

$$x+iy=aw^m+O(|w|^{m+1}),$$

$$z=\operatorname{Re}(bw^{m+n})+O(|w|^{m+n+1}),\quad w=u+iv,$$

where $a=c(1+i)\neq0$ and $b\neq0$ are two complex constants, $\operatorname{Im}c=0$; $m\geq2$, and $n\geq1$ are integers named, respectively, the order and the index of the singular point and $u$ and $v$ are intrinsic isothermal coordinates.

The following theorem was deduced on the basis of this representation: If the numbers $m+n$ and $m$ are coprime, then the minimal surface has $(m-1)(m+n)$ different lines of self-intersection issuing from the singular point in different directions. There is a relation between the genus of a complete minimal surface, the number of its branching points, and the index of its Gaussian mapping [1].

One distinguishes between two kinds of branching points: false branching points and true (non-false) branching points. False branching points are singularities of the mapping which defines the surface that can be got rid of by re-parametrization (e.g. if $r=r(w)$ is a regular minimal surface, then the generalized minimal surface $r=r(w^2)$ has a false branching point at $w=0$). A true branching point represents a real singularity of the surface itself, and has the following important property: In a neighbourhood of a true branching point the surface can be altered so that the new surface which coincides with the original one outside the deformed neighbourhood will have a smaller area than the original surface (this holds for surfaces in $\mathbf R^3$ but is not true in more-general settings: e.g. for the area-minimizing surfaces in $\mathbf R^4$). The theory of generalized minimal surfaces with a branching point served as a base of the general theory of branched immersions, developed for a broad class of two-dimensional surfaces in $\mathbf R^n$, $n\geq3$, [2].

References

[1]	Y.W. Chen, "Branch points, poles and planar points of minimal surfaces in $\mathbf R^3$" Ann. of Math. , 49 : 4 (1948) pp. 790–806 MR0028086 Zbl 0038.33102
[2]	J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1973) MR0448224 Zbl 0319.53003