Stable minimal surfaces in $\textbf {R}^4$ with degenerate Gauss map

Abstract:

A complete oriented stable minimal surface in $\textbf {R}^3$ is a plane, but in $\textbf {R}^4$, there are many non-flat examples such as holomorphic curves. The Gauss map plays an important role in the theory of minimal surfaces. In this paper, we prove that a complete oriented stable minimal surface in $\textbf {R}^4$ with $\alpha$-degenerate Gauss map (for $\alpha > 1/4$) is a plane.