Tok srednje zakrivljenosti ploha u euklidskom prostoru

master's thesis

• Title (english): The flow of mean surface curvature in Euclidean space
• Author: Ivan Krznarić
• Mentor: Željka Milin Šipuš (mentor)
• Committee member: Željka Milin Šipuš (predsjednik povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-07-14, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilj ovog rada jest obraditi osnove teorije toka srednje zakrivljenosti hiperploha u euklidskom prostoru. U prvom poglavlju rada dan je pregled osnovnih pojmova i rezultata teorije ravninskih krivulja. Dalje, drugo je poglavlje posvećeno toku skraćivanja ravninskih krivulja. Najprije je definiran pojam toka skraćivanja u obliku određene Cauchyjeve zadaće, zatim je dokazan teorem o egzistenciji i jedinstvenosti rješenja toka te su proučena najbitnija svojstva toka. Također, dokazan je Gage-Hamiltonov teorem koji u potpunosti opisuje tok konveksnih krivulja te je naveden Graysonov teorem koji pak u potpunosti opisuje tok jednostavno zatvorenih krivulja. U trećem poglavlju posvetili smo se geometriji hiperploha u euklidskom prostoru, uveli smo osnovne pojmove diferencijalnog i tenzorskog računa na hiperplohama te smo definirali pojam srednje zakrivljenosti. U četvrtom poglavlju bavili smo se tokom srednje zakrivljenosti hiperploha u euklidskom prostoru. Tok srednje zakrivljenosti uveli smo u obliku Cauchyjeve zadaće, dokazali smo egzistenciju i jedinstvenost rješenja te smo dokazali neka osnovna svojstva toka poput principa izbjegavanja. Na kraju poglavlja, naveden je Huiskenov teorem kojim smo u potpunosti opisali tok kompaktnih konveksnih hiperploha.

Abstract (english)

Goal of this paper is to examine the basic properties of the mean curvature flow of hypersurfaces in Euclidean space. In the first chapter, a brief review of the theory of plane curves is given. In the second chapter, the curve shortening flow is analyzed. Firstly, we defined the curve shortening flow in the form of a Cauchy problem, then the existence and uniqueness theorem for the solution is obtained and the main properties of the flow are studied. Furthermore, we proved the Gage-Hamilton theorem which gives a complete description of the curve shortening flow of convex curves and we showed how Grayson's theorem gives a full description of the curve shortening flow for simply closed curves. The third chapter is used for the review of the basic results regarding the differential and tensor calculus on hypersurfaces as well as the definition of the mean curvature. Finally, in the fourth chapter we introduced the mean curvature flow of a hypersurface in the form of a Cauchy problem. We proved the existence and uniqueness theorem for the solution of the flow as well as some of the basic properties of the flow, for example the avoidance principle. In the end of the chapter, we stated Huisken's theorem which gives a complete description of the mean curvature flow of compact convex hypersurfaces.

• Keywords: zakrivljenosti hiperploha; teorija ravninskih krivulja; Cauchyjeve zadaće; Gage-Hamiltonov teorem; Graysonov teorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:909693
• Study programme: Title: Theoretical Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-02-09 11:46:54