Unicity theorems for the Gauss maps of complete minimal surfaces. II
In Part I [J. Math. Soc. Japan 45, No. 3, 481-487 (1993; Zbl 0796.53005)] the author proved a uniqueness theorem for the Gauss maps of complete minimal surfaces in \(\mathbb{R}^3\). In the present paper the author gives some generalizations to minimal surfaces in \(\mathbb{R}^m\) in the case \(m > 3\). Recall that the Gauss map of a minimal surface \(M\) in \(\mathbb{R}^m\) can be thought of as a holomorphic map of \(M\) into the complex projective space \(\mathbb{C} P^{m - 1}\). Suppose we have two minimal surfaces \(M\) and \(\overline {M}\) with Gauss maps \(G : M \to \mathbb{C} P^N\), \(\overline {G} : \overline {M} \to \mathbb{C} P^N\), where \(N = m - 1\). If \(\Phi : M \to \overline {M}\) is a conformal diffeomorphism, we assume that the holomorphic maps \(f = G\) and \(g = \overline {G} \circ \Phi\) satisfy the following: There exist hyperplanes \(H_1, \dots, H_q\) in \(\mathbb{C} P^N\) in general position such that (i) \(f^{-1} (H_j) = g^{-1} (H_j) \neq M\) for all \(j\), (ii) \(f = g\) on \(\bigcup^q_{j = 1} f^{-1} (H_j) - K\) for a compact subset \(K\) of \(M\). The author proves that \(f \equiv g\) in the following two cases: (A) if \(q > m^2 + m(m - 1)/2\), \(M\) is complete and has infinite total curvature; (B) if \(q \geq m^2 + m(m - 1)/2\), \(K = \emptyset\) and \(M\), \(\overline {M}\) are both complete and have finite total curvature.
scientific article; zbMATH DE number 558882
scientific article