On the Gauss curvature of minimal surfaces
In the present paper, the author improves one of his results [J. Differ. Geom. 29, No. 2, 245-262 (1989; Zbl 0676.53005)] as follows: Let \(x=(x_ 1,x_ 2,x_ 3):M\to\mathbb{R}^ 3\) be a minimal surface immersed in \(\mathbb{R}^ 3\) and let \(G: M\to S^ 2\) be the Gauss map of \(M\). Assume that \(G\) omits five distinct unit vectors \(n_ 1,\dots,n_ 5\in S^ 2\). Let \(\theta_{ij}\) be the angle between \(n_ i\) and \(n_ j\) and set \(L:=\min\{\sin\bigl({\theta_{ij}\over 2}\bigr)\); \(1\leq i < j\leq 5\}\). Then there exists some positive constant \(C\) not depending on each minimal surface such that \[ | K(p)|^{1/2}\leq {C\over d(p)}{\log^ 2(1/L)\over L^ 3},\qquad p\in M \] .
scientific article; zbMATH DE number 98456