Plohe konstantne srednje zakrivljenosti

master's thesis

• Author: Petra Vukašinović
• Mentor: Željka Milin-Šipuš (mentor)
• Committee member: Željka Milin-Šipuš (predsjednik povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Committee member: Goran Igaly (član povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Teorija ploha konstantne srednje zakrivljenosti je veoma komplicirano područje matematike koje zahtijeva poznavanje diferencijalne geometrije i kompleksne analize te smo stoga jedno cijelo poglavlje posvetili upravo osnovama istih koje bi čitatelj ovog rada trebao razumjeti. U drugom smo se poglavlju fokusirali na temu ovog rada i dali definiciju ploha konstantne srednje zakrivljenosti kao plohe čija je srednja zakrivljenost jednaka nekoj konstanti. Kako ta konstanta može biti i nula, posebno smo razmatrali minimalne plohe, odnosno plohe čija je srednja zakrivljenost jednaka nuli i plohe čija je srednja zakrivljenost različita od nule. Za minimalne plohe smo dali primjere, Weierstrass-Enneperovu reprezentaciju te pokazali kako se katenoid može deformirati u helikoid pri čemu su sve plohe deformacije također minimalne. Na kraju smo izveli Kenmotsuovo rješenje, odnosno uvjete koji moraju biti ispunjeni da bi neka rotacijska ploha bila konstantne srednje zakrivljenosti te smo razmatrali Delaunayev zaključak da je neka ploha rotacijska ploha konstantne srednje zakrivljenosti ako i samo ako je njena profilna krivulja ruleta konike.

Abstract (english)

The theory of surfaces of constant mean curvature is a very complex area of mathematics, which requires understanding of differential geometry and complex analysis. Therefore, we dedicate an entire chapter to the basics of these mathematical branches, which need to be understood by the reader. The second chapter focuses on the subject of this dissertation. We define surfaces of constant mean curvature as surfaces whose mean curvatures equal a constant. Seeing as this constant may also be zero, we separately approach minimal surfaces, defined as those surfaces that have a mean curvature value of zero, and surfaces with a non-zero value of mean curvature. We provide examples and a Weierstrass-Enneper representation of minimal surfaces, and we show how a catenoid may be deformed into a helicoid, where all the surfaces in the associated family of a catenoid are also minimal. Finally, we derive Kenmotsu’s solution, that is to say we define the conditions which need to be met for a surface of revolution to have a constant mean curvature. We also discuss Delaunay’s conclusion that a surface of revolution is a surface of constant mean curvature if and only if its profile curve is a roulette of a conic.

• Keywords: plohe konstantne srednje zakrivljenosti; minimalne plohe; Weierstrass-Enneperova reprezentacija; katenoid; helikoid; Kenmotsuovo rješenje; Delaunayev zaključak
• Keywords (english): surfaces of constant mean curvature; minimal surfaces; Weierstrass-Enneper representation; catenoid; helicoid; Kenmotsu’s solution; Delaunay’s conclusion
• Language: croatian
• URN:NBN: urn:nbn:hr:217:887528
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-02-19 16:36:45