Fusheng Wei’s doubly periodic surface of genus 2 is remarkable for both its historical significance and the amount of research it has triggered.
It was the first genus 2 embedded doubly periodic minimal surface, adding a handle to one of the Karcher-Meeks-Rosenberg examples. It comes in a 1-parameter family (which is part of a larger, 3-dimensional family). Extreme cases are shown below.
The surface on the left is close to a noded limit where the catenoidal nodes are periodically placed at the vertices of rhombi whose diagonals have ratio
Limits like these are well understood thanks to work of Martin Traizet.
The other limit looks like two Scherk surfaces glued together using catenoidal necks as glue. This limit is not understood at all yet.
After the classification of complete, embedded doubly periodic minimal tori by Rodríguez-Pérez-Traizet, the genus 2 case offers a challenge. At this point, we don’t even know whether the space of 4-ended examples is connected.
F. Wei: Some existence and uniqueness theorems for doubly
periodic minimal surfaces, Invent. Math 109 (1992), 113–136.
- Higher Genus Wei Surfaces
- Wei’s Genus 4 Triply Periodic Surface