Plohe konstantne srednje zakrivljenosti

The theory of surfaces of constant mean curvature is a very complex area of mathematics, which requires understanding of differential geometry and complex analysis. Therefore, we dedicate an entire chapter to the basics of these mathematical branches, which need to be understood by the reader. The second chapter focuses on the subject of this dissertation. We define surfaces of constant mean curvature as surfaces whose mean curvatures equal a constant. Seeing as this constant may also be zero, we separately approach minimal surfaces, defined as those surfaces that have a mean curvature value of zero, and surfaces with a non-zero value of mean curvature. We provide examples and a Weierstrass-Enneper representation of minimal surfaces, and we show how a catenoid may be deformed into a helicoid, where all the surfaces in the associated family of a catenoid are also minimal. Finally, we derive Kenmotsu’s solution, that is to say we define the conditions which need to be met for a surface of revolution to have a constant mean curvature. We also discuss Delaunay’s conclusion that a surface of revolution is a surface of constant mean curvature if and only if its profile curve is a roulette of a conic.