Exploring minimal surfaces modulo p

Project leader: Professor Demetre Kazaras
Project manager: Kai Xu
Team members: Hiba Benjeddou, Ben Goldstein, Khiyali Pillalamarri, Nico Salazar

A minimal surface is a surface in 3-space which locally minimizes surface area. These structures are found in nature as membranes experiencing equal pressure from opposing sides, such as soap films spanning a wireframe. In our project we investigated and found new examples of 'minimal mod-3 surfaces'. These objects are made up of pieces of minimal surfaces meeting each other 3 at a time along 'singular curves' so that the angles between colliding faces are always 120 degrees. Not much is known about these minimal mod-3 surfaces -- our goal was to find new examples and investigate what properties the singular curves might have.


DOmath 2022 Kazaras project image file 1


We first considered surfaces of revolution, obtained by rotating appropriate collections of catenary curves. Infinitely many examples were found, and their singular curves display a rich structure. Another class was constructed with a 'screw-motion symmetry' by carefully arranging helicoidal surfaces to intersect at 120 degrees. Our third approach was to design wire-frames with special symmetry so that the minimal-area surface spanning them could be extended to all of 3-space by rotation and translation. We also investigated the intrinsic geometry of the mod-3 surfaces themselves, understanding how distances behave on a particular example called the 'round mod-3 disc'.


DOmath 2022 Kazaras project image file 2