Minimalne ploskve
magistrsko delo
Abstract
Konformna imerzija iz odprte Riemannove ploskve v Evklidski prostor ${\mathbb R}^{n}$, $n \geq 3$, je minimalna natanko tedaj, ko je harmonična. Ta osnovni pogoj karakterizira minimalne ploskve, ki so po definiciji stacionarne točke ploskovnega funkcionala. Najpreprostejša primera katenoida in helikoid, znana že v 18. stoletju, nastaneta kot realni in imaginarni del holomorfne ničelne krivulje helikatenoide. Ideja aproksimacije in interpolacije minimalnih ploskev, osrednje teme magistrskega dela, so klasični izreki za holomorfne funkcije. Periodno dominantni spreji, Morsejeva teorija in teorija konveksne integracije Gromova o obstoju poti s predpisanimi integrali nam omogočajo iskanje bližnjih preslikav z ničelnimi realnimi periodami, ki po Enneper-Weierstrassovi formuli določajo minimalne ploskve. Izkaže se, da izreki tipa Mergelyana, Weierstrassa in Mittag-Lefflerja veljajo za konformne minimalne imerzije ter splošnejše holomorfne ničelne krivulje, pri čemer v obeh primerih lahko izberemo prave preslikave.
Keywords
matematika;minimalne ploskve;Riemannove ploskve;konformne harmonične preslikave;Rungejev izrek;Weierstrassov izrek;
Data
| Language: | Slovenian |
|---|---|
| Year of publishing: | 2022 |
| Typology: | 2.09 - Master's Thesis |
| Organization: | UL FMF - Faculty of Mathematics and Physics |
| Publisher: | [T. Vrhovnik] |
| UDC: | 517.5 |
| COBISS: |
93868035
|
| Views: | 1389 |
| Downloads: | 181 |
| Average score: | 0 (0 votes) |
| Metadata: |
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Other data
| Secondary language: | English |
|---|---|
| Secondary title: | Minimal surfaces |
| Secondary abstract: | A conformal minimal immersion from an open Riemann surface into the Euclidean space ${\mathbb R}^{n}$, $n \geq 3$, is minimal if and only if it is harmonic. This fundamental condition characterizes minimal surfaces, formally defined as stationary points of the area functional. The simplest examples, known since the 18th century, are catenoid and helicoid, the real and imaginary parts of the holomorphic null curve called helicatenoid. The idea behind approximation and interpolation of minimal surfaces, our main goal, are classical theorems for holomorphic functions, although they need to be suitably adapted. Period dominating sprays, Morse theory and Gromov’s convex integration theory concerning the existence of paths with prescribed integrals enable us to find nearby maps with vanishing real periods, which define minimal surfaces by the Enneper-Weierstrass formula. It turns out that theorems of Mergelyan, Weierstrass and Mittag-Leffler type hold for conformal minimal immersions as well as more general holomorphic null curves. Additionally, such immersions can be chosen to be proper. |
| Secondary keywords: | mathematics;minimal surfaces;Riemann surfaces;conformal harmonic maps;Runge theorem;Weierstrass theorem; |
| Type (COBISS): | Master's thesis/paper |
| Study programme: | 0 |
| Thesis comment: | Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 2. stopnja |
| Pages: | IX, 67 str. |
| ID: | 14305966 |
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