The soul curve of cmc tori
Final Report Abstract
The main objective of this project is to prove a conjecture of Ulrich Pinkall, which roughly says that there are enough tori of constant mean curvature (cmc tori) in the three-dimensional euclidean space to approximate an arbitrary closed curve. This conjecture is the result of the introduction of new tools to geometric analysis which started with Henry Wente’s discovery of the first cmc torus in three dimensional euclidean space 45 years ago. This discovery itself disproved the persuasion of Heinz Hopf that there do not exist such cmc tori at all. The new tools apply a correspondence between complex curves and solutions of PDEs which are related to so-called integrable systems. The main scientific innovation of the project consists of the introduction of deformations which transition the boundaries between different integrable systems related to cmc tori and closed curves, respectively. The transition has been divided into two successive transitions, which are both related to so called blow-ups. Here a blow-up is a modern mathematical tool and applies to both sides of the correspondence, the complex curves and the solutions of PDEs. Roughly speaking it has been understood how to transition from the cmc tori to the closed curves. What is still missing is the reverse of this transition, and to produce for a given curve a sequence of cmc tori approximating the given curve. The good news is that we found no obstacle to the success of the whole approach.
Publications
- On closed finite gap curves in space forms II, Journal of Integrable Systems 5 (2020), xyaa002, 25 pages
S. Klein, M. Kilian
(See online at https://doi.org/10.1093/integr/xyaa002) - The blow-up limit of sequences of constant mean curvature tori with fixed spectral genus in the euclidean 3-space
E. Carberry, S. Klein, M.U. Schmidt
(See online at https://doi.org/10.48550/arXiv.2110.01574) - The closure of the space of spectral curves of constant mean curvature tori in R3 with spectral genus 2
E. Carberry, M. Kilian, S. Klein, M.U. Schmidt
(See online at https://doi.org/10.48550/arXiv.2110.00436)
