Plohe konstantne srednje zakrivljenosti i njima pridružene fokalne krivulje i plohe u Minkowskijevom prostoru

doctoral thesis

• Title (english): Constant mean curvature surfaces and associated focal curves and surfaces in Minkowski space
• Author: Ivana Protrka
• Mentor: Željka Milin-Šipuš (mentor)
• Committee member: Zlatko Erjavec (predsjednik povjerenstva)
• Committee member: Juraj Šiftar (član povjerenstva)
• Committee member: Zdenka Kolar-Begović (član povjerenstva)
• Committee member: Ema Jurkin (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

U radu su proučavane plohe konstantne srednje zakrivljenosti (CMC plohe) i njima pridružene fokalne krivulje, odnosno plohe u Minkowskijevom prostoru. Minkowskijev prostor je specijalan ambijentni prostor u kojemu, s obzirom na definiranu pseudo-metriku, razlikujemo tri vrste vektora, krivulja, odnosno ploha. U prva tri poglavlja (1, 2, 3) opisani su osnovni pojmovi koje koristimo u radnji te su prikazana osnovna svojstva, primjeri i klasifikacija minimalnih, maksimalnih i pravih CMC ploha u Minkowskijevom prostoru. U poglavlju 4 proučavana je \(\alpha\)-evoluta i njen poseban slučaj, harmonijska evoluta plohe. U prvom dijelu istraživana su svojstva harmonijskih evoluta pravčastih, helikoidalnih i translacijskih ploha. Dobiveni rezultati su analogni rezultatima u euklidskom prostoru. U drugom dijelu proučavana su svojstva harmonijskih evoluta kvazi-umbiličkih ploha. Kod takvih ploha harmonijska evoluta degenerira u krivulju što se ne može dogoditi u euklidskom prostoru. Štoviše, dokazano je da harmonijska evoluta i bazna krivulja B-namotajne plohe konstantne srednje zakrivljenosti (\(H = const. \neq 0\)) čine svjetlosni Bertrandov par. Nadalje, pokazano je kako od zadane krivulje konstruirati B-namotajnu plohu čija je harmonijska evoluta zadana krivulja. Na kraju poglavlja istraživana su svojstva \(\alpha\)-evolute, a dobiveni rezultati predstavljaju znatna proširenja rezultata u euklidskom prostoru. U poglavlju 5 proučavane su cikloidne krivulje. Izvedena je njihova parametrizacija i pokazano je da se iste mogu dobiti kao projekcije sfernih općih zavojnica. Konstruirani su novi primjeri maksimalnih ploha pomoću Björlingove formule, tzv. cikloidne maksimalne plohe. Dan je njihov Weierstrassov prikaz i pokazano je da su dobivene plohe algebarske te da dopuštaju racionalnu parametrizaciju.

Abstract (english)

In this doctoral thesis, the surfaces of constant mean curvature (CMC surfaces) and associated focal sets in Minkowski space are studied. Minkowski space is a special ambient space in which, with respect to defined pseudo-metric, we distinguish three types of vectors, curves and surfaces. In the first three chapters of the thesis, we give an overview of basic concepts and theorems in Minkowski space, local theory of curves and surfaces, as well as examples and classification of minimal, maximal and CMC surfaces in Minkowski space. In the chapter 4, the \(\alpha\)-evolute and its special case, the harmonic evolute of a surface, are analyzed. In the first part, we studied harmonic evolute of ruled, helicoidal and translation surfaces, respectively. Obtained results are analogous to the results in Euclidean case. In the second part, we studied the harmonic evolute of the quasi-umbilic surfaces. Contrary to the results in Euclidean case, the harmonic evolute of such surfaces is a curve. Moreover, it is proved that the harmonic evolute of a B-scroll of constant non-vanishing mean curvature, together with its base curve forms a null Bertrand pair. Furthermore, it is shown how to reconstruct B-scroll from a given harmonic evolute. At the end of the chapter, we studied properties of \(\alpha\)-evolute and the obtained results present significant extension of the results in Euclidean case. In the chapter 5, the cycloidal curves are considered. Their parametrization is presented, as well as shown that they can be obtained as projections of spherical generalized helices. By using the Björling formula, new examples of maximal surfaces are constructed, so-called cycloidal maximal surfaces. We derived their Weierstrass representations and proved that given surfaces are algebraic surfaces which admit rational parametrization.

• Keywords: Minkowskijev prostor; minimalne plohe; maksimalne plohe; CMC plohe; α-evoluta; \(\alpha\)-evoluta; alfa-evoluta; harmonijska evoluta; kvazi-umbiličke plohe; cikloidne krivulje; cikloidne maksimalne plohe
• Keywords (english): Minkowski space; minimal surfaces; maximal surfaces; CMC surfaces; α-evolute; \(\alpha\)-evolute; alpha-evolute; harmonic evolute; quasi-umbilical surface; cycloidal curves; cycloidal maximal surfaces
• Language: croatian
• URN:NBN: urn:nbn:hr:217:938710
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: vi, 167 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-11-29 10:14:38