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Minimal Graphs in Riemannian Fibrations

Alex, Tristan (2016):
Minimal Graphs in Riemannian Fibrations.
Darmstadt, Technische Universität Darmstadt, [Online-Edition: http://tuprints.ulb.tu-darmstadt.de/5571],
[Ph.D. Thesis]

Abstract

In this dissertation, minimal and constant mean curvature surface theory in 3-dimensional Riemannian fibrations are studied.

In the main part of the thesis, new complete, embedded minimal surfaces in the 3-sphere are constructed by solving a Plateau problem with respect to a suitable Jordan curve consisting entirely of horizontal geodesic arcs and extending this solution by means of Schwarz reflection.

Additionally, an elementary proof for the vertical half-space theorem in Heisenberg space is given by finding a subsolution of the minimal surface equation.

Finally, projections of constant mean curvature multigraphs are characterized: they are locally contained to one side of complete curves with constant geodesic curvature.

Item Type: Ph.D. Thesis
Erschienen: 2016
Creators: Alex, Tristan
Title: Minimal Graphs in Riemannian Fibrations
Language: English
Abstract:

In this dissertation, minimal and constant mean curvature surface theory in 3-dimensional Riemannian fibrations are studied.

In the main part of the thesis, new complete, embedded minimal surfaces in the 3-sphere are constructed by solving a Plateau problem with respect to a suitable Jordan curve consisting entirely of horizontal geodesic arcs and extending this solution by means of Schwarz reflection.

Additionally, an elementary proof for the vertical half-space theorem in Heisenberg space is given by finding a subsolution of the minimal surface equation.

Finally, projections of constant mean curvature multigraphs are characterized: they are locally contained to one side of complete curves with constant geodesic curvature.

Place of Publication: Darmstadt
Uncontrolled Keywords: Minimal surfaces, constant mean curvature surfaces, maximum principle, Plateau, reflection principles, Riemannian fibration, model geometry
Divisions: 04 Department of Mathematics > Applied Geometry
04 Department of Mathematics
Date Deposited: 17 Jul 2016 19:55
Official URL: http://tuprints.ulb.tu-darmstadt.de/5571
URN: urn:nbn:de:tuda-tuprints-55719
Referees: Große-Brauckmann, Prof. Dr. Karsten and Fröhlich, Prof. Dr. Steffen
Refereed / Verteidigung / mdl. Prüfung: 6 July 2016
Alternative Abstract:
Alternative abstract Language
In dieser Dissertation werden Minimalflächen und Flächen konstanter mittlerer Krümmung in dreidimensionalen Riemannschen Faserungen studiert. Im Hauptteil der Arbeit werden neue vollständige, eingebettete Minimalflächen in der 3-Sphäre konstruiert, indem zunächst ein Plateauproblem bezüglich einer geeigneten Jordankurve gelöst wird, die nur aus horizontalen Geodätischen besteht. Die Plateaulösung wird mit dem Schwarz-Spiegelungsprinzip zu einer vollständigen, eingebetteten Fläche fortgesetzt. Zusätzlich wird ein elementarer Beweis für den vertikalen Halbraumsatz im Heisenbergraum angegeben, indem eine konkrete Sublösung für die Minimalflächengleichung gefunden wird. Im letzten Teil werden die Projektionen von Multigraphen konstanter mittlerer Krümmung charakterisiert: sie liegen lokal auf einer Seite von Kurven konstanter geodätischer Krümmung.German
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