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# Minimal and constant mean curvature surfaces in various three -manifolds

#### Abstract

This work is divided into three sections. In the first, we construct new complete finite total curvature minimal surfaces in $\IR\sp3$ with ends asymptotic to catenoids. These have positive genus and maintain the symmetries of their genus zero counterparts. We use the conjugate surface construction, convergence arguments, and continuity arguments in the proofs. The next section is concerned with the construction of constant mean curvature surfaces in $\IR\sp3$. We construct CMC surfaces with symmetries of the rectangle and with embedded Delaunay ends. These ends alternate between ones asymptotic to nearly spherical Delaunay surfaces to ones asymptotic to nearly cylindrical Delaunay surfaces. We also construct continuous families a Delaunay towers, again ones nearly spherical and ones nearly cylindrical. We establish the existence of the CMC patches by proving the existence of minimal patches in $S\sp3$, it process referred to as cousin conjugate construction. Convergence and continuity arguments are also used. In the final section, we construct minimal surfaces of every genus in $S\sp2\times S\sp1$. We directly solve Plateau's problem for fundamental boundary curves, and then extend via geodesic reflection to the complete surfaces. ^

Mathematics

#### Recommended Citation

Jan-Olof Jorgen Berglund, "Minimal and constant mean curvature surfaces in various three -manifolds" (January 1, 1996). Doctoral Dissertations Available from Proquest. Paper AAI9721432.
http://scholarworks.umass.edu/dissertations/AAI9721432

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