A rigidity theorem for the minimal surfaces of \(E^ 3\) (Q1902496)

A minimal surface \(f : M^2 \to \mathbb{R}^3\) is called minimally rigid if any other minimal immersion \(g : M^2 \to \mathbb{R}^3\) isometric to \(f\) is congruent to \(f\). In \textit{H. I. Choi, W. H. Meeks III} and \textit{B. White}, J. Differ. Geom. 32, No. 1, 65-76 (1990; Zbl 0704.53008)] it was proved that if \(f\) is properly embedded and has more than one end, then \(f\) is minimally rigid. In the present paper, the authors reformulate the main lemma of the above paper and prove the following result. Let \(f : M^2 \to \mathbb{R}^3\) be a minimal immersion. Assume that \(M^2\) is orientable and that its Gaussian curvature is not identically zero. Then \(f\) is minimally rigid if and only if \((*df)\) is exact; here \((*df)\) is the Hodge dual of \(df\) (which takes values in \(\mathbb{R}^3\)). The case of nonorientable surfaces is treated directly and they are shown to be minimally rigid. Some applications are given and some examples are discussed.