Isometric immersions of tori into R³ with given mean curvature

Abstract

It is known that a compact Riemannian surface can admit at most two (2) geometrically distinct, i. e., non-congruent isometric immersions into R ³ with given non-constant mean curvature. If the genus is zero, then there is at most one such immersion. Here, we show that there is at most one such immersion in each of the following cases for surfaces of genus one: 1) there exists a point p such that (H ² − K)(p) = 0, where K is the curvature of the Riemannian metric and H is the given non-constant mean curvature (umbilic point); 2) the surface is a surface of revolution; 3) the surface is a tube formed by moving a circle in such a way that its center describes a smooth plane curve and its plane is constantly perpendicular to this curve. We also indicate the difficulties as to why the so-far existing methodologies cannot give a clear-cut answer to the question if it is possible to reduce the at most two immersions to at most one for any compact Riemannian surface of genus greater than zero.