On uniqueness of Riemann’s examples

Abstract

We prove that a properly embedded minimal annulus with one flat end, bounded in a slab by lines or circles, is a part of a Riemann’s example. In 1956 Shiffman [16] proved that a minimal annulus bounded by a pair of circles in two parallel planes intersects every plane between the two planes by a circle. Therefore, by a result of Riemann and Enneper ([12], pages 85-90), it must be a part of the catenoid or a part of a Riemann’s example. In [7], Hoffman, Karcher and Rosenberg proved that a properly embedded minimal annulus bounded by two parallel lines in a slab must be a piece of a Riemann’s example. Toubiana [17] proved that no properly embedded minimal annulus in a slab can be bounded by a pair of nonparallel lines. In [5] the first author generalized these results to a properly embedded minimal annulus in a slab bounded by a combination of circles or lines in parallel planes. He showed that the annulus must be a part of a catenoid or a part of a Riemann’s example; thus if the boundary consists of two lines, the lines must be parallel. In this paper, we generalize the first author’s result to allow the minimal annulus to have a flat end in the slab. Denote S = {(x1, x2, x3) : −1 ≤ x3 ≤ 1} and let S−1 and S1 be the two boundary planes of S at x3 = −1, 1, respectively. We prove the following theorem. Theorem 1. Suppose A ⊂ S is a properly embedded minimal annulus with a flat end. If ∂A consists of a line or circle in S−1 and a line or circle in S1, then A intersects every plane between S−1 and S1 by a circle, except at the height of the end, where the intersection is a line. Consequently, A is a piece of a Riemann’s example. In particular, if the boundary consists of two lines, the lines must be parallel. Some remarks are in order before we proceed to the proof. It is known that Riemann’s examples are properly embedded minimal annuli with infinitely many flat ends. Any slab parallel to the ends cut such a surface by an annulus with a finite number of ends. It is natural to ask whether or not Riemann’s examples are the only properly embedded minimal annuli with infinitely many flat ends, or whether they are the only properly embedded minimal annuli in a slab with a finite number of ends. The above theorem only deals with annuli in a slab with one end. Prior to it, Romon [15] proved the same result under the assumption that the Received by the editors October 21, 1996. 1991 Mathematics Subject Classification. Primary 53A10; Secondary 35P99. The first author is supported by the Australian Research Council. c ©1998 American Mathematical Society