Topology and curvature of minimal surfaces properly embedded in \(\mathbb{R}^ 3\) (Q676101)

The author studies the connection between the topology and the total curvature \(C(M)\) of minimal surfaces properly embedded in \(\mathbb{R}^3\). The main result is: Let \(E\), which is homeomorphic to \(S^1\times [0,+\infty]\), be a ring-shaped minimal surface properly embedded in the half-space \(H= \{(x_1,x_2,x_3) \in\mathbb{R}^3 \mid x_3 >0)\}\), intersecting every horizontal plane in a Jordan curve and let \(\partial E\) be a smooth curve of \(\partial H\). Then \(E\) has finite total curvature. From this theorem follow two results: Theorem 1. Every ring-shaped handle \(E\) of a minimal surface \(M\) properly embedded in \(\mathbb{R}^3\), which has at least two handles, is of finite total curvature. Theorem 2. Let \(M\) be a minimal surface properly embedded in \(\mathbb{R}^3\) and let \(M\) have at least two handles. Then \(M\) has finite topology if and only if \(M\) has finite total curvature.