Constant mean curvature surfaces of higher genus

• Applicants: Privatdozent Dr. Sebastian Heller; Dr. Nicholas Schmitt
• Subject Area: Mathematics
• Term: from 2014 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 251913428

• Final Report Year: 2019

Final Report Abstract

A systematic integrable systems approach to CMC surfaces of higher genus in the 3-sphere (and in euclidean 3-space) was lacking since the early 90s, when the theory for CMC tori has been established . First results in this direction have been obtained by the principal investigator S.H.. The aim of the project was to deepen the understanding of the integrable structure of CMC surfaces of higher genus, to provide new examples of surfaces (both theoretically and experimentally), to develop new tools for the investigation of CMC surfaces and to exploit similarities and differences to the case of CMC surfaces of genus 1. We succeeded in obtaining these goals. We developed a generalised Whitham deformation theory which enabled us to construct new examples of (branched) CMC surfaces of higher genus. The deformation theory yields a first picture of how the structure of the moduli space of CMC surfaces of higher genus might look like. In this respect, it also highlights the main similarities and differences to (a part of) the moduli space of CMC tori. The theoretical investigations have been supplemented by computer experiments, which led to fascinating (experimental) insights about the structure of the space of CMC surfaces of higher genus in the 3-sphere. The experiments gave strong evidence to the existence of new CMC surfaces of higher genus. In the meantime, the idea of deforming the associated family of gauge classes has become popular in the surface geometry community, and has also been applied in different research areas. We plan to apply the developed methods in other contexts and to sharpen the techniques in future work.

Publications

• Exploring the space of symmetric CMC surfaces
  L. Heller, S. Heller, N. Schmitt
  
• The spectral curve theory for (k, l)−symmetric CMC surfaces, J. Geom. Phys., Volume 98 (2015), pp. 201–213
  L. Heller, S. Heller, N. Schmitt
  
• Abelianization of Fuchsian Systems on a 4-punctured sphere and applications, Journal of Symplectic Geometry, Vol. 14, No. 4 (2016), pp. 1059–1088
  L. Heller, S. Heller
  
• The asymptotic behavior of the monodromy representations of the associated families of compact CMC surfaces, Bull. London Math. Soc. (2016) 48 (5), 729–734
  S. Heller
  
• Automorphisms of a rank one Deligne- Hitchin moduli space , SIGMA 13 (2017), 072, 19 pages
  I. Biswas, S. Heller
  
• Navigating the Space of Symmetric CMC Surfaces, Journal of Differential Geometry, Vol. 110, No. 3 (2018), pp. 413–455
  L. Heller, S. Heller, N. Schmitt