An estimate for the Gaussian curvature of minimal surfaces in \(\mathbb R^m\) whose Gauss map is ramified over a set of hyperplanes (Q389358)

A map \(f = (f_0:\dots:f_{m-1}) : \mathbb C \to \mathbb P^{m-1}(\mathbb C)\) is said to be ramified over a hyperplane \(H = \{(w_0:\dots:w_{m-1}) \in \mathbb P^{m-1}(\mathbb C) | a_0w_0 + \dots + a_{m-1}w_{m-1} = 0\}\) with multiplicity at least \(e\) if all zeros of the function \((f,H):=a_0f_0 + \dots + a_{m-1}f_{m-1}\) have orders at least \(e\). If \(f\) omits \(H\), then \(f\) is ramified with multiplicity equal to \(\infty\). Using this notion, in the paper the author proves the estimate NEWLINE\[NEWLINE|K(p)|^{{1}/{2}}d(p) \leq CNEWLINE\]NEWLINE for the Gaussian curvature \(K(p)\) of a minimal surface \(M \subset \mathbb R^{m}\) whose Gauss map is ramified over a set of hyperplanes with multiplicities satisfying some inequality. In the estimate above \(d(p)\) stands for the distance from \(p \in M\) to the boundary of \(M\), and the constant \(C\) depends only on the set of hyperplanes, but not on the surface. If \(M\) is complete, this result yields flatness of the surface and constancy of its Gauss map. The main theorem of the paper generalizes previously known results due to A. Ros, R. Osserman, M. Ru, and others. The proof combines techniques of Osserman and Ru with some results from the geometric orbifold theory.