Minimalne ploskve : delo diplomskega seminarja

Gačnik, Katarina (author), Kuzman, Uroš (mentor)

V delu diplomskega seminarja obravnavamo minimalne ploskve. To je, dvodimenzionalne objekte, katerih površina je lokalno minimalna. S pomočjo variacijskega računa bomo izpeljali Euler-Lagrangeovo parcialno diferencialno enačbo, ki ji mora zadoščati vsaka eksplicitno podana minimalna ploskev. Nadalje bomo pokazali, da je parametrično podana ploskev minimalna natanko tedaj, ko ima ničelno srednjo ukrivljenost. Nazadnje si bomo ogledali še primer, ki potrdi, da minimalne ploskve niso nujno tudi globalno ekstremne. To pomeni, da lahko pri danih robnih pogojih najdemo več minimalnih ploskev z različnimi površinami.

• Language: Slovenian
• Keywords: minimalne ploskve, lokalni in globalni minimum, variacijski račun, diferencialna geometrija
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2018
• PID: 20.500.12556/RUL-102146
• UDC: 514.7
• COBISS.SI-ID: 18411865
• Publication date in RUL: 20.07.2018

Secondary language

• Language: English
• Title: Minimal surfaces

Abstract

In this dissertation we study minimal surfaces. That is, two-dimensional objects whose area is locally minimal. Using the calculus of variations we will derive the Euler-Lagrange differential equation which has to be fulfilled for explicitly given minimal surfaces. Further, we will show that a parametric surface is minimal if and only if its mean curvature equals zero. Finally, we will present an example which points out that a minimal surface is not always a global extreme. This means that, given boundary conditions, there may exist several minimal surfaces with different areas.

• Keywords: minimal surfaces, local and global minimum, calculus of variations, differential geometry