论文标题
最小表面，最大表面，及时的最小表面和出生的式孤子的有限分解
Finite Decomposition of Minimal surfaces, Maximal surfaces, Timelike Minimal surfaces and Born-Infeld solitons
论文作者
论文摘要
我们表明，使用Euler Ramanujan身份，Scherk的第二表面的高度函数分解为自身的缩放和翻译版本的有限总和。 R. Kamien在液晶上的工作中出现了类似的结果，他在其中显示（使用Euler-Ramanujan身份），Scherk的第一个表面分解为自身的缩放和翻译版本的有限总和。我们在翻译后的螺旋内表面和翻译后的第一个表面进行了另一个有限的分解，以施尔克的第一个表面的高度功能。我们提供了更多示例，例如（复杂的）最大表面和（复杂的）bi soliton。 We then show, using the Weierstrass-Enneper representation of minimal (maximal) surfaces, that one can decompose the height function of a minimal (maximal) surface into finite sums of height functions of surfaces which, upon change of coordinates, turn out to be minimal (maximal) surfaces, each minimal (maximal) w.r.t.到自己的新坐标。然后，我们展示了最小表面，最大表面，及时的最小表面和出生的污染孤子表面的一般特性，即它们的局部高度功能$ z = z = z（x，y）$分为相同形式的缩放和翻译版本的有限总和。缩放这些新功能是最小表面，最大表面，及时的最小表面和出生的式孤子表面的高度功能。最后，我们展示了$ {\ mathbb r}^3 $的叶面，通过移动的螺旋可以（出现在Euler-Ramanujan的一个身份）中。
We show that the height function of Scherk's second surface decomposes into a finite sum of scaled and translated versions of itself, using an Euler Ramanujan identity. A similar result appears in R. Kamien's work on liquid crystals where he shows (using an Euler-Ramanujan identity) that the Scherk's first surface decomposes into a finite sum of scaled and translated versions of itself. We give another finite decomposition of the height function of the Scherk's first surface in terms of translated helicoids and scaled and translated Scherk's first surface. We give some more examples, for instance a (complex) maximal surface and a (complex) BI soliton. We then show, using the Weierstrass-Enneper representation of minimal (maximal) surfaces, that one can decompose the height function of a minimal (maximal) surface into finite sums of height functions of surfaces which, upon change of coordinates, turn out to be minimal (maximal) surfaces, each minimal (maximal) w.r.t. to its own new coordinates. We then exhibit a general property of minimal surfaces, maximal surfaces, timelike minimal surfaces and Born-Infeld soliton surfaces that their local height functions $z=Z(x,y)$ split into finite sum of scaled and translated versions of functions of the same form. Upto scaling these new functions are height functions of the minimal surfaces, maximal surfaces, timelike minimal surfaces and Born-Infeld soliton surfaces respectively. Lastly, we exhibit a foliation of ${\mathbb R}^3$ minus certain lines by shifted helicoids (which appear in one of the Euler-Ramanujan identities).
