ON SURFACES OF STATIONARY AREA BOUNDED BY TWO CIRCLES, OR CONVEX CURVES, IN PARALLEL PLANES*
ON SURFACES OF STATIONARY AREA BOUNDED BY TWO CIRCLES, OR CONVEX CURVES, IN PARALLEL PLANES*
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DOI：
10.2307/1969991
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发表时间：
1956
影响因子：
4.9
通讯作者：
M. Shiffman
中科院分区：
文献类型：
--
作者：
M. Shiffman
垂直于连接它们中心的直线的。问题来了，这是否是仅有的两个被两个圆包围的最小曲面。答案是肯定的，因此我们有明确的例子表明空间中的曲线包围了不止一个最小曲面，而且它们都是已知的。更一般地说，我们还将考虑两个边界圆位于平行平面但不一定有共同对称轴的情况。我们将证明，任何以它们为界的最小曲面都具有这样的性质，即它与两个边界圆的平面平行的平面相交，仍然是一个圆。这一性质使得以两个圆为界的最小曲面可以明确地确定，事实上，B. Riemann已经在这个假设下做出了这一确定[6，第305-352页，特别是第341-347页]。精确地说，设S是空间中由平行平面上的两条平面曲线In， r2所包围的最小曲面。以下定理将被证明：定理1。如果Fi， F2是圆，那么S与F1， F2平行的平面的交点也是圆。定理2。如果ri r2是凸曲线，S与平行于rF平面的平面相交，r2也是凸曲线。参考定理1，很容易得出，当两个圆有一个共同的对称轴时，那么所有以ri， r2为界的最小曲面都是旋转对称的，因此是链状面。这完成了一个经典的例子，并提供了一个边界曲线包围多个已知最小曲面的例子。在这方面，应该提到T. Rado在1946年普林斯顿200周年纪念会议上提出的问题：估计以某条曲线为界的最小曲面的数目。2. 设S是用保形参数（u, v）表示的最小曲面。它由方程x = x(u, v) y = y(u, v) z = z（u, v）给出，其中函数x(u, v) y(u, v) z（u, v）是（u, v）和x？+ yu + zu = xv + y？+ zV, +YYv + zuZv = 0。
pendicular to the straight line joining their centers. The question presents itself whether these are the only minimal surfaces bounded by the two circles. The answer is in the affirmative, and we therefore have explicit examples of curves in space bounding more than one minimal surface and all of them known. More generally, we shall also consider the case when the two bounding circles lie in parallel planes but do not necessarily have a common axis of symmetry. It will be shown that any minimal surface bounded by them has the property that its intersection by a plane parallel to the planes of the two bounding circles is again a circle. This property allows the minimal surfaces bounded by the two circles to be explicitly determined and indeed this determination has already been made by B. Riemann under this assumption [6, pp. 305-352, especially pp. 341-347]. In precise terms, let S be a minimal surface in space bounded by two plane curves IN, r2 lying in parallel planes. The following theorems will be proved: THEOREM 1. If Fi, F2 are circles, then the intersection of S by a plane parallel to the planes of F1, F2 is again a circle. THEOREM 2. If ri, r2 are convex curves, the the intersection of S by a plane parallel to the planes of rF, r2 is again a convex curve. With reference to Theorem 1, it easily follows that when the two circles have a common axis of symmetry, then all the minimal surfaces bounded by ri, r2 are rotationally symmetric and are therefore catenoids. This completes a classical example, and furnishes an example of boundary curves bounding more than one minimal surface all of which are known. In this connection, mention should be made of the question posed by T. Rado in the Princeton Bicentennial Conference in 1946: to estimate the number of minimal surfaces bounded by a given curve or curves. 2. Let S be a minimal surface given in terms of conformal parameters (u, v). It is given by the equations x = x(u, v), y = y(u, v), z = z(u, v), where the functions x(u, v), y(u, v), z(u, v) are harmonic functions of (u, v) and x? + yu + zu = xv + y? + zV, +YYv + zuZv = O.