Topics in complex approximation theory : doctoral thesis

Chenoweth, Brett Simon (author), Forstnerič, Franc (mentor)

In this thesis, we investigate problems in complex approximation theory motivated by recent developments in Oka theory, minimal surface theory, and contact geometry. Primarily, our focus lies in proving approximation results in the spirit of Carleman’s theorem, that is, better than uniform approximation on noncompact sets. The original research of this dissertation begins in Chapter 3, where we prove a generalisation of Carleman’s theorem for maps from Stein manifolds to Oka manifolds. Then, in Chapter 4, we prove a version of Carleman’s theorem for directed holomorphic immersions and minimal surfaces. Under suitable hypotheses, we may even ensure that the approximating maps have desirable global properties, including completeness and properness. As an application of these results, we give an approximate solution to a Plateau problem for divergent Jordan curves in Euclidean spaces. Finally, Chapter 5 is concerned with approximation by solutions of systems of differential equations. We adapt the tools and techniques that have successfully been applied in the single equation, contact case. Period dominating sprays play an instrumental role.

• Language: English
• Keywords: Stein manifold, Oka manifold, holomorphic map, Carleman approximation, bounded exhaustion hulls, minimal surface, directed holomorphic curve
• Work type: Doctoral dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• PID: 20.500.12556/RUL-121510
• Publication date in RUL: 13.10.2020

Secondary language

• Language: Slovenian
• Title: Teme v kompleksni aproksimacijski teoriji

Abstract

V disertaciji obravnavamo probleme v kompleksni aproksimacijski teoriji, ki so motivirani s teorijo Oka, teorijo minimalnih ploskev in holomorfno kontaktno geometrijo. Delo je osredotočeno na aproksimacijske rezultate Carlemanovega tipa, to je aproksimacijo v fini topologiji na nekompaktnih zaprtih množicah. Originalni rezultati disertacije se pričnejo v poglavju 3 z dokazom posplošitve Carlemanovega izreka za preslikave Steinovih mnogoterosti v mnogoterosti Oka. V poglavju 4 je dokazana verzija Carlemanovega izreka za usmerjene holomorfne imerzije in konformne minimalne imerzije. Ob ustreznih predpostavkah lahko zagotovimo dodatne lastnosti aproksimantov kot so kompletnost in pravost. Kot primer uporabe dobljenih rezultatov dokažemo obstoj približnih rešitev Plateaujevega problema za divergentne Jordanove krivulje v Evklidskih prostorih. V poglavju 5 obravnavamo aproksimacijo rešitev sistemov holomorfnih diferencialnih enačb z uporabo metod, nedavno razvitih za aproksimacijo Legendrovih krivulj v kompleksnih kontaktnih mnogoterostih.

• Keywords: Steinova mnogoterost, Oka mnogoterost, holomorfna preslikava, Carlemanova aproksimacija, omejena ogrinjača, minimalna ploskev, usmerjena holomorfna krivulja