CMC Tori in the Generalised Berger Spheres and their Duals
(2022) In Bachelor’s Theses in Mathematical Sciences MATK11 20221Mathematics (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:
http://lup.lub.lu.se/student-papers/record/9088489
- author
- Gegenfurtner, Johanna Marie LU
- supervisor
- organization
- course
- MATK11 20221
- year
- 2022
- 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}}, }