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Constant mean curvature surfaces in Euclidean and hyperbolic 3-space
Abstract
This dissertation consists of three parts. The first part is an assortment of results about the geometry, topology, stability, existence, and limiting behavior of compact minimal surfaces with a pair of planar boundary curves in Euclidean 3-space $\IR\sp3$. The second part classifies all $C\sp2$ foliations by proper complete finite topology constant mean curvature surfaces of 3-manifolds, where the 3-manifolds are $\IR\sp3$ or hyperbolic 3-space $\IH\sp3$ with an isolated collection of curves and points removed. Such foliations in $\IR\sp3$ consist only of planes, cylinders, and spheres. In $\IH\sp3$ we assume every leaf has mean curvature at least as great as that of a horosphere, and we conclude such foliations consist only of horospheres, geodesic spheres, and hyperbolic Delaunay surfaces. The third part considers limit surfaces of sequences of Riemann's examples in $\IR\sp3$ (converging in the $C\sp\infty$-topology in compact regions) and concludes that the attainable limits are precisely (1) a single plane; (2) an infinite number of equally spaced parallel planes; (3) a Riemann example; (4) a catenoid; and (5) a helicoid.
Subject Area
Mathematics
Recommended Citation
Rossman, Wayne F, "Constant mean curvature surfaces in Euclidean and hyperbolic 3-space" (1992). Doctoral Dissertations Available from Proquest. AAI9305888.
https://scholarworks.umass.edu/dissertations/AAI9305888