A characterization of Riemann's minimal surfaces
Riemann's minimal surfaces in \(\mathbb R ^3\) are surfaces of genus zero with infinite number of planar ends that form a one-parametric family of complete, embedded, singly-periodic minimal surfaces foliated by circles and lines in parallel planes. As shown by Riemann, they are the only minimal surfaces apart from the catenoid that are fibered by circles in parallel planes. If \(T\) is the group generated by a nontrivial translation in \(\mathbb R ^3,\) then Riemann's minimal surfaces can be considered as minimal tori of genus one with two planar ends in \(\mathbb R ^3/T.\) The authors' main result is that Riemann's minimal surfaces are the only properly embedded minimal tori with two planar ends in \(\mathbb R ^3/T.\) In fact, the authors find all the complete immersed minimal tori with two parallel embedded planar ends, among which only Riemann's examples are actually embedded. The used technique consists of an analysis of the periods as holomorphic functions of the conformal structures first used by \textit{F. J. López} [Trans. Am. Math. Soc. 334, 49-74 (1992; Zbl 0771.53005)]. There is a conjecture due to W. H. Meeks and H. Rosenberg that Riemann's examples together with the plane, the catenoid and the helicoid, are the only properly embedded minimal surfaces of genus zero in \(\mathbb R ^3.\) A somewhat weaker conjecture can be formulated in \(\mathbb R ^3/T\) to the effect that Riemann's examples are the only properly embedded minimal surfaces of genus one with even number of ends in \(\mathbb R ^3/T.\) The above result of the paper establishes the validity of this conjecture when the number of ends is two, that is the smallest one.
scientific article; zbMATH DE number 1145697