The first eigenvalue of Laplacians on minimal surfaces in \(S^ 3\) (Q760016)

There are many complete surfaces with constant mean curvature in the Euclidean 3-space \(R^ 3\) and in the hyperbolic 3-space \(H^ 3\) [see \textit{K. Kenmotsu}, Tôhuku Math. J., II. Ser. 32, 147-153 (1980; Zbl 0431.53005), \textit{H. Mori}, Trans. Am. Math. Soc. 278, 671-687 (1983; Zbl 0518.53011)]. But in the Euclidean 3-sphere \(S^ 3\) there have been few results on such surfaces except umbilic ones and flat tori [cf. \textit{K. Nomizu} and \textit{B. Smyth}, J. Differ. Geom. 3, 367-377 (1969; Zbl 0196.251)]. In this paper, we shall construct a one-parameter family of complete, rotational surfaces in \(S^ 3\) with constant mean curvature, including a flat torus as an initial one. In particular, there is a one- parameter family of complete, rotational, minimal surfaces in \(S^ 3\), including the Clifford torus. And we shall show that non of the closed, rotational, minimal surfaces in \(S^ 3\) is embedded and the first eigenvalues of some ones relative to the Laplacian are smaller than two except for the Clifford torus.