Topics in complex approximation theory

doctoral dissertation

Brett Chenoweth (Author), Franc Forstnerič (Mentor)

Abstract

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.

Keywords

Stein manifolds;Oka manifolds;holomorphic maps;Carleman approximation;bounded exhaustion hulls;minimal surfaces;directed holomorphic curves;

Data

Language:	English
Year of publishing:	2020
Typology:	2.08 - Doctoral Dissertation
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[B. Chenoweth]
UDC:	514.752:517.55(043.3)
COBISS:	32648963
Views:	1365
Downloads:	127
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Teme v kompleksni aproksimacijski teoriji
Secondary 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.
Secondary keywords:	Steinove mnogoterosti;Oka mnogoterosti;holomorfne preslikave;Carlemanova aproksimacija;omejena ogrinjača;minimalne ploskve;usmerjene holomorfne krivulje;
Type (COBISS):	Doctoral dissertation
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. Ljubljana, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 3. stopnja
Pages:	VIII, 120 str.
ID:	12074666