Separable minimal surfaces and their limit behavior (Q6619749)

The theory of minimal surfaces in a given space is a classical subject in differential geometry. If \(\Sigma\) is a regular surface in the Euclidean space \(\mathbb{R}^3\), then \(\Sigma\) is said to be separable if \[\Sigma=\lbrace (x, y, z)\in\mathbb{R}^3, f(x)+g(y)+h(z)=0\rbrace\] where \(f\), \(g\) and \(h\) are real-valued functions. In this paper, the authors fully categorize all singly, doubly, and triply periodic minimal surfaces and they give precise equations for separable minimal surfaces with elliptic functions. The main theorem of this article is a classification of separable minimal surfaces in \(\mathbb{R}^3\) with an implicit form \(f(x) + g(y) + h(z) = 0\).