Geometrija minimalnih ploskev : delo diplomskega seminarja

Derin, Tanja (author), Černe, Miran (mentor)

In this thesis, we present minimal surfaces and from where the interest in them comes. We start with the definition of the area functional and find its stationary point, a minimal surface. Consequently, we get a necessary condition for minimal surfaces i.e., to have the mean curvature equaling zero everywhere. In the next section, we connect the field of minimal surfaces with complex analysis. With the introduction of the Enneper-Weierstrass representation formula, we get a tool for generating minimal surfaces by choosing an appropriate holomorphic and a meromorphic function. In the last section, we study the Björling’s problem to find a minimal surfrace containing the given curve with the prescribed normal. We prove that the solution can be uniquely written with a formula.

• Language: Slovenian
• Keywords: Minimalna ploskev, kompleksna analiza, konformna mapa, Björlingov problem, Enneper-Weierstrass reprezentacija
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2023
• PID: 20.500.12556/RUL-148247
• UDC: 514.7
• COBISS.SI-ID: 162137091
• Publication date in RUL: 05.08.2023

Secondary language

• Language: English
• Title: Geometry of minimal surfaces

Abstract

V diplomski nalogi bomo predstavili, kaj so minimalne ploskve in iz kje izvira zanimanje za te. Izhajali bomo iz definicije ploskovnega funkcionala, iz katerega dobimo osnovno karakterizacijo minimalnih ploskev in sicer, so to ploskve z ničelno povprečno ukrivljenostjo. V naslednjem razdelku pa z znanjem iz kompleksne analize izpeljemo Enneper-Weierstrassovo formulo, ki nam omogoči z ustrezno izbiro ene holomorfne in druge meromorfne funkcije generiranje minimalnih ploskev. V zadnjem razdelku omenimo Björlingov problem, katerega zahteva je, da najademo minimalno ploskev, ki bo vsebovala dano krivuljo in katere bo normala sovpadala z normalo krivulje. Dokažemo, da rešitev tega problema obstaja in je natanko določena s predpisom.

• Keywords: minimal surface, complex analysis, conformal map, the Björling’s problem, Enneper-Weierstrass representation