A Note on Shiffman's Theorems |
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Authors: | Yi Fang Jenn-Fang Hwang |
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Institution: | (1) Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT, 0200, Australia;(2) Institute of Mathematics, Academia Sinica, Nankang, Taipei, 11529, Taiwan Republic of China |
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Abstract: | Shiffman proved his famous first theorem, that if A R3 is a compact minimal annulus bounded by two convex Jordan curves in parallel (say horizontal) planes, then A is foliated by strictly 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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Keywords: | minimal surfaces convex curves curvature |
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