Gap theorems for minimal submanifolds of Euclidean space (Q1087156)

The author studies complete minimal submanifolds M properly immersed into Euclidean space \({\mathbb{R}}^ N\). Suppose that the scalar curvature \[ (*)\quad R\geq -A/(1+| x|^{2+\epsilon}) \] for some positive constants A and \(\epsilon\). Then: M is an m-plane if \(m=\dim M\geq 3\) and M has one end. Since an area-minimizing hypersurface M properly embedded into \({\mathbb{R}}^ N\) has one end, M is a hyperplane provided that M satisfies condition (*). Similar results hold for \(m=2:\) M is a plane if \(\epsilon\geq 2\) and M has one end, or if \(\epsilon >2\) and M is embedded into \({\mathbb{R}}^ 3\). Enneper's surface and the catenoid satisfy condition (*) with \(\epsilon =2/3\) resp. \(\epsilon =2\).