Constant Mean Curvature Surfaces bifurcating from Nodoids.
Technische Universität, Darmstadt
[Ph.D. Thesis], (2012)
Available under Creative Commons Attribution Non-commercial No Derivatives, 2.5.
Download (5MB) | Preview
|Item Type:||Ph.D. Thesis|
|Title:||Constant Mean Curvature Surfaces bifurcating from Nodoids|
In this work we construct families of CMC (Constant Mean Curvature) surfaces which bifurcate from certain well-known rotational surfaces. The elementary principle of construction in doing so is the Lawson's correspondence which establishes a 1 to 1 relation between simply connected minimal surfaces in one space form of curvature K and the isometric CMC surfaces with mean curvature c in another space form with curvature K-c^2. Two different cases are discussed in this work. In the first case we construct new CMC surfaces, which bifurcate from the immersed rotational CMC surfaces in the 3-dimensional Euclidean space, namely the nodoids. Mazzeo und Pacard have showed a local (i.e. near nodoids) existence of such surfaces. In this work we shall use conjugate surfaces method to construct the complete family of bifurcating CMC surfaces to the point of degeneration. In the second case we shall construct a 1-parameter family of single periodic minimal surfaces which bifurcating from the helicoid. According to its scope the work was divided into two parts. In part 1 we introduce the boundary arcs (geodesic quadrilateral) of the fundamental patch in the 3-sphere. The Plateauproblems to the new quadrilaterals are solvable. To guarantee the regularity of the Plateau solution in extending by the Schwarz reflection across its geodesic boundary arcs we use the covering solid cylinder of the solid Clifford torus and hemi-sphere as barriers. We generalize the argument of Rado for the 3-sphere and with that we show that the Plateau solution is a Graph over the 2-sphere in a Hopf fibration. It follows a uniqueness result for the Plateau solution and the continuity of the bifurcation family. The new singly periodic CMC surfaces are immersed 2-sphere with two punctures and has discrete symmetry. In part 2 we shall apply the conjugate surfaces method for hyperbolic CMC-1 surfaces to construct family of bifurcating minimal surfaces from helicoid in Euclidean space. The key issue here is the solution of Plateau problems for non-compact boundary curves. The deformation parameter is the surface normal at infinity. By a exhaustion process using compact minimal surfaces we get the existence of a non-compact minimal surface. A standard curvature estimate for minimal surface equation yields the desired asymptotic property of the normal vector. The new minimal surfaces are singly periodic and constitute a 1-parameter family.
|Place of Publication:||Darmstadt|
|Classification DDC:||500 Naturwissenschaften und Mathematik > 510 Mathematik|
|Divisions:||04 Department of Mathematics > Applied Geometry|
|Date Deposited:||31 Jan 2012 10:51|
|Last Modified:||07 Dec 2012 12:04|
|Referees:||Große-Brauckmann, Prof. Dr. Karsten and Tomi, Prof. Dr. Friedrich|
|Refereed:||9 February 2010|