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CMC Tori in the Generalised Berger Spheres and their Duals

Gegenfurtner, Johanna Marie LU (2022) In Bachelor’s Theses in Mathematical Sciences MATK11 20221
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
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 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... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
Gegenfurtner, Johanna Marie LU
supervisor
organization
course
MATK11 20221
year
type
M2 - Bachelor Degree
subject
keywords
Constant mean curvature, Berger Spheres, Lorentzian Geometry, Minimal surfaces, Differential Geometry
publication/series
Bachelor’s Theses in Mathematical Sciences
report number
LUNFMA-4135-2022
ISSN
1654-6229
other publication id
2022:K8
language
English
id
9088489
date added to LUP
2025-06-27 15:51:07
date last changed
2025-06-27 15:51:07
@misc{9088489,
  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 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.}},
  author       = {{Gegenfurtner, Johanna Marie}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor’s Theses in Mathematical Sciences}},
  title        = {{CMC Tori in the Generalised Berger Spheres and their Duals}},
  year         = {{2022}},
}