Helicoidal Minimal Surfaces


These pages survey results obtained by Martin Traizet and myself, see our preprint.

The study of minimal surfaces with helicoidal ends was originally motivated by the desire to construct a genus g helicoid as the limit of screw-motion invariant minimal surfaces with genus g in the quotient for increasing twist angle. While this limit is hard, it is easier to understand the limits of such surfaces for decreasing twist angle.We expect that these surfaces must degenerate in a very peculiar way. In fact, by starting at certain degenerate situations, we can prove the existence of minimal surface families that are invariant under screw motions and have helicoidal ends.

We start with a finite set of points in the plane. These points come positively or negatively charged. To the right you see a symetric configuration of five points, four of them negatively charged. Meeks' Möbius strip
These points must satisfy a set of balance equations: Lopez' Klein Bottle
Then we consider a non-minimal surface which is the graph of the following multi-valued function: Slab Surface  
The result is a negatively curves surface in space with vertical axes through the points of the configuration, and sometimes horizontal lines when the configuration is symmetric. Slab Surface