A note on Shiffman's theorems

Research Report MRR98-034

A note on Shiffman's theorems

Yi Fang and Jenn-Fang Hwang

Abstract: Shiffman in [10] proved his famous first theorem, that if A \subset R³ is a compact minimal annulus bounded by two convex Jordan curves in parallel (say horizontal) planes, then A is foliated by convex horizontal Jordan curves. In this article we use Perron's method to construct minimal annuli which have a planar end and are bounded by two convex Jordan curves in horizontal planes, but the horizontal level sets of the surfaces are not all convex Jordan curves or straight lines. These surfaces show that unlike his second and third theorems, Shiffman's first theorem is not generalizable without further qualification.

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