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On convexity of level curves. (English) Zbl 0697.53016

Let M be a minimal surface in space bounded by two plane curves \(C_ 1\), \(C_ 2\) lying in parallel planes. If \(C_ 1\), \(C_ 2\) are convex curves, then the intersection of M by a plane parallel to the planes of \(C_ 1\), \(C_ 2\) is again a convex curve. That assertion has been established by M. Shiffman [Ann. Math., II. Ser. 63, 77-90 (1956; Zbl 0070.168)]. In this note the author shows that, by constructing a counterexample, the above assertion is false for surfaces with constant non-zero mean curvature.
Reviewer: T.Hasanis

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53A05 Surfaces in Euclidean and related spaces

Citations:

Zbl 0070.168
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