Minimal surfaces bounded by convex curves in parallel planes
Minimal surfaces bounded by convex curves in parallel planes
复制标题
由平行平面中的凸曲线界定的最小曲面
DOI：
10.1007/bf02566647
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发表时间：
1991
影响因子：
0.9
通讯作者：
B. White
中科院分区：
文献类型：
--
作者：
W. Meeks;B. White
1956年M. Shiffman [17]证明了几个关于极小环A的几何的美丽定理，该极小环A的边界由平行平面P1，P2中的两条闭凸曲线组成。第一个定理指出，A与任何平面P的交点，在P1和P2之间，是凸约当曲线。特别地，可以得出A是嵌入的。然后，他用这个凸性定理证明，A的边界的每一个对称性都可以扩展到A的一个对称性。在这种情况下，A包括两个圆圈Shiffman证明，A是foliated圆在平行的平面。早些时候的B。黎曼[15]描述，在椭圆函数，所有最小环3，可以表示为联盟的圆在平行的平面（也见[3]为一个
In 1956 M. Shiffman [17] proved several beautiful theorems concerning the geometry of a minimal annulus A whose boundary consists of two closed convex curves in parallel planes P1, P2. The first theorem stated that the intersection of A with any plane P , between P1 and P2, is a convex Jordan curve. In particular it follows that A is embedded. He then used this convexity theorem to prove that every symmetry of the boundary of A extended to a symmetry of A. In the case that ∂A consists of two circles Shiffman proved that A was foliated by circles in parallel planes. Earlier B. Riemann [15] described, in terms of elliptic functions, all minimal annuli in 3 that can be expressed as the union of circles in parallel planes (also see [3] for a