CMC Tori in the Generalised Berger Spheres and their Duals

Abstract

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 applications 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 $S^3$ and its dual space $\Sigma^3$. 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 $S^3$, computing its Levi-Civita connection and sectional curvatures: in Chapter 2 with respect to the Riemannian metric g and in Chapter 4 with respect to the Lorentzian metric h. Further, we determine some minimal and CMC tori inside ($S^3$, g) in Chapter 3 and in ($S^3$, h) in Chapter 5. We then proceed with the dual space $\Sigma^3$ of $S^3$. 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 ($\Sigma^3$, g) in Chapter 7 and in ($\Sigma^3$, h) in Chapter 9. In the appendix, the reader will find a Maple program. It was written to check the computations of the $S^3$ cases, but it can easily be adapted to$\Sigma^3$.