Stable minimal surfaces in 𝑅⁴ with degenerate Gauss map

Shoda, Toshihiro

doi:10.1090/S0002-9939-03-07332-5

Stable minimal surfaces in $\textbf {R}^4$ with degenerate Gauss map
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Proc. Amer. Math. Soc. 132 (2004), 1285-1293 Request permission

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.

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