SOME BASIC FACTS, OLD AND NEW, ABOUT TRIPLY PERIODIC EMBEDDED MINIMAL SURFACES
SOME BASIC FACTS, OLD AND NEW, ABOUT TRIPLY PERIODIC EMBEDDED MINIMAL SURFACES
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关于三周期嵌入极小曲面的一些新旧基本事实
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发表时间：
1990
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影响因子：
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通讯作者：
D. Hoffman
中科院分区：
文献类型：
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作者：
D. Hoffman
在过去的二十年中，三周期嵌入最小表面已经引起了化学、晶体学和材料科学研究人员的极大兴趣和应用。Schwarz， Neovious等人的经典例子可以追溯到19世纪，但似乎停留在现代数学对最小曲面的兴趣的边缘。舍恩的专著，可以说是现代人对这门学科的兴趣之源，提出了许多新的例子，但科学家对这项工作的了解远远超过数学家。除了陀螺仪，直到最近，舍恩的例子在数学领域才有了很大的影响。最近，由于Karcher、Fischer-Koch、Wohlgemuth、Nitsche和Ross的研究，这种情况有所改变。当然，科学家们最感兴趣的是，这些三周期嵌入的最小表面是两个相互交织在一起的固体区域的共同边界，被称为迷宫，它们本身就是三周期的。这些迷宫被定义为表面补体的组成部分。最小曲面具有非正高斯曲率，这似乎是迷宫界面的一个重要的次要性质。这样的表面使得迷宫的底层对称群变得非常明显。在这些最小曲面在科学上的一些应用中，它们平均曲率为零的事实似乎并不是一个关键性质。在这些情况下，分割面最小的假设可能是一个令人困惑和不必要的限制。为了说明这一点，请考虑以下关于三周期嵌入最小曲面的问题列表。
In the last twenty years, triply-periodic embedded minimal surfaces have been of great interest and utility to researchers in chemistry, cystallography and material science. The classical examples of Schwarz, Neovious et al., dating from the 19th century were known, but seemed to rest on the periphery of modern mathematical interest in minimal surfaces. The monograph of Schoen, which may fairly be said to be the source of modern interest in the subject, presented many new examples, but this work was much better known to scientists than to mathematicians. Except for the gyroid, few of Schoen’s examples have had very much impact, until quite recently, in the field of mathematics proper. This has changed recently due in part to the work of Karcher, Fischer-Koch, Wohlgemuth, Nitsche, and Ross. Of course, of primary interest to scientists is the fact that these triply-periodic embedded minimal surfaces are the common boundary of two connected intertwined solid regions, referred to as labyrinths that are themselves triply-periodic. These labyrinths are defined as the components of the complement of the surface in . Minimal surfaces have nonpositive Gauss curvature and this seems to be an important secondary property of the labyrinthine interface. Such a surface makes very plain the underlying symmetry group of the labyrinths. In some of the applications of these minimal surfaces in science, the fact that they have zero mean curvature does not seem to be a critical property. In those cases, the assumption that the dividing surface is minimal may be a confusing and unnecessary restriction. To illustrate this, consider the following list of questions about triply-periodic embedded minimal surfaces.