Extrinsic diameter of immersed flat tori in \(S^3\)
The Clifford torus in \(S^{3}\) is defined as \(M_{\theta}=\{(z,w)\in \mathbb{C}^{2}:|z|^{2}=\cos^{2}\theta,|w|^{2}=\sin^{2}\theta\},0<\theta<\frac{\pi}{2}\). It is a flat embedded torus of constant mean curvature. Denote by \(i_{\theta}:M_{\theta}\rightarrow S^{3}\) the canonical inclusion and by \(ds_{\theta}^{2}\) the induced metric on \(M_{\theta}\). This paper proves the following. If \(f:(M_{\theta},ds_{\theta}^{2})\rightarrow S^{3}\) is an isometric immersion, and if the mean curvature function \(H\) of \(f\) does not change sign on \(M_{\theta}\) , then \(f\) is congruent to \(i_{\theta}\) . The proof of this depends on the technique obtained by \textit{M. Umehara} [Nagoya Math. J. 134, 75--89 (1994; Zbl 0806.53002)]. A \(C^{\infty}\) map \(f:M^{n}\rightarrow N^{n+1}\) is called a wave front or front if \(f\) lifts to a Legendrian immersion \(L_{f}:M^{n}\rightarrow P(T^{\ast} N^{n+1})\), where \(P(T^{\ast} N^{n+1})\) is the projectified cotangent bundle of \(N^{n+1}\). A representation formula was obtained for flat surfaces in \(S^{3}\), regarded as wave fronts and this generalizes the first author's correspondence result [J. Math. Soc. Japan 40, No.3, 457--476 (1988; Zbl 0642.53059)]. Also, they provide two applications, one for the existence of a flat torus, a wave front with extrinsic diameter less than \(\pi\) and the other one for the sufficient and necessary condition for when two given tori can be deformed into each other within the class of flat fronts.
scientific article; zbMATH DE number 6101878
scientific article