CMC Tori in the Generalised Berger Spheres and their Duals - Gegenfurtner,Johanna Marie (Bachelor’s Theses in Mathematical Sciences, 2022)
The study of minimal surfaces has a long history, due to the important applications. Given a fixed boundary, one wants to minimise the surface area: this can be used, for example, to minimise the area of the roof of a building. Similarly, looking for constant mean curvature (CMC) provides us with many interesting ap- plications in physics – one of the easiest examples are soap bubbles. In this work however we occupy ourselves with minimal and constant mean curvature surfaces in the three-dimensional sphere S3 and its dual space Σ3. In Chapter 1 we give a brief overview of the tools of Riemannian and Lorentzian geometry that we will use. We then take a closer look at S3, computing its Levi- Civita connection and sectional curvatures: in Chapter 2 with respect to the Rie- mannian metric g and in Chapter 4 with respect to the Lorentzian metric h. Further, we determine some minimal and CMC tori inside (S3,g) in Chapter 3 and in (S3, h) in Chapter 5. We then proceed with the dual space Σ3 of S3. In Chapter 6, we calculate the Levi-Civita connection and sectional curvatures with respect to g, and with respect to h in Chapter 8. Again we look for minimal and CMC tori of a certain family in (Σ3, g) in Chapter 7 and in (Σ3, h) in Chapter 9. In the appendix, the reader will find a Maple program. It was written to check the computations of the S3 cases, but it can easily be adapted to Σ3.
Keywords: Constant mean curvature; Berger Spheres; Lorentzian Geometry; Minimal surfaces; Differential Geometry