Periodic Mean Curvature Research Articles

Rotational hypersurfaces of periodic mean curvature

In this paper we discuss rotational hypersurfaces in R n and more specifically rotational hypersurfaces with periodic mean curvature function. We show that, for a given real analytic function H ( s ) on R , every rotational hypersurface M in R n with mean curvature H ( s ) can be extended infinitely in the sense that all coordinate functions of the generating curve of M are defined on all of R as well. For rotational hypersurfaces with periodic mean curvature we present a criterion characterizing the periodicity of such hypersurfaces in terms of their mean curvature function. We also discuss a method to produce families of periodic rotational hypersurfaces where each member of the family has the same mean curvature function. In fact, given any closed planar curve with curvature κ, we prove that there is a family of periodic rotational hypersurfaces such that the mean curvature of each element of the family is explicitly determined by κ. Delaunay's famous result for surfaces of revolution with constant mean curvature is included here as the case where n = 3 and κ is constant.

Surfaces of revolution with periodic mean curvature

The surfaces of revolution with constant mean curvature in 3 are classified by Delaunay [2] in 1841. They are locally plane, catenoid, sphere, circular cylinder, unduloid, and nodoid up to isometries of 3. On these 10 years, new and interesting examples of non-zero constant mean curvature surfaces are discovered. In the global study of complete surfaces with constant mean curvature, unduloids and nodoids play important role as the models of ends of such surfaces (see [5], [6]). The work by Delaunay is now revived after 150 years of his discovery. The purpose of this paper is to study surfaces of revolution with periodic mean curvature in order to extend the theory of constant mean curvature surfaces. In general such a surface is not periodic, because the catenoid gives the counter-example. First we show the criterion for a periodic function to be the mean curvature of a periodic surface of revolution and second describe a method how to construct these periodic surfaces of revolution whose mean curvatures are periodic functions satisfying the criterion. I thank Professor Yusuke Sakane for his interest and plotting of the beautiful pictures by the computer of Osaka University which I used in the seminar talk of Granada University on March, 2001, and in the AMS meeting held in Hoboken on April, 2001. The figures of this paper are all programmed by the author using the symbolic manipulation program Mathematica. I also thank my graduate student Shinya Hirakawa for his help to simplify the programing.