CMC Tori in the Generalised Berger Spheres and their Duals

TL;DR

This paper investigates minimal and constant mean curvature tori within the three-dimensional sphere and its dual space, providing detailed geometric computations and classifications in both Riemannian and Lorentzian settings.

Contribution

It offers new explicit examples and classifications of CMC and minimal tori in $S^3$ and its dual $\,\Sigma^3$, including detailed curvature calculations and a computational Maple tool.

Findings

Explicit minimal and CMC tori identified in $S^3$ and $\,\Sigma^3$

Curvature properties of these spaces computed in Riemannian and Lorentzian metrics

A Maple program for verifying and extending the geometric computations

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$\Sigma^3$.