View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b0175201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b0175202.png" />, over the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b0175203.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b0175204.png" /> the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b0175205.png" /> of the minimal surface can be represented in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b0175206.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b0175207.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b0175208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b0175209.png" /> are two complex constants, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752010.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752011.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752012.png" /> are integers named, respectively, the order and the index of the singular point and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752014.png" /> are intrinsic isothermal coordinates.

The following theorem was deduced on the basis of this representation: If the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752016.png" /> are coprime, then the minimal surface has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752017.png" /> 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 [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752018.png" /> is a regular minimal surface, then the generalized minimal surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752019.png" /> has a false branching point at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752020.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752021.png" /> but is not true in more-general settings: e.g. for the area-minimizing surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752022.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752024.png" />, [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> Y.W. Chen, "Branch points, poles and planar points of minimal surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017520/b01752025.png" />" ''Ann. of Math.'' , '''49''' : 4 (1948) pp. 790–806 {{MR|0028086}} {{ZBL|0038.33102}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1973) {{MR|0448224}} {{ZBL|0319.53003}} </TD></TR></table>