Minimal submanifolds in \(\mathbb{S}^{p-3}\) with constant scalar curvature
Let \(S^n\) denote an \(n\)-dimensional unit sphere. In this paper, the authors prove: Theorem 1: Let \(M\subset S^{3+p}\), \(p\geq 1\), be a closed 3-dimensional minimal submanifold with a flat normal bundle and constant scalar curvature \(R\geq 0\). Then \(R= 0\), \(R=3\) or \(R=6\). Theorem 2: Let \(x:M\to S^{3+p}\) be a substantial isometric immersion. Suppose in addition that \(M\) is a closed 3-dimensional minimal submanifold with a flat normal bundle and constant scalar curvature \(R\leq 0\). Then \(p\leq 2\) and \(M\) is (a) an equatorial 3-sphere \(S^3\), (b) a Clifford torus \(S^1(\sqrt 3/3)\times S(\sqrt 6/3)\), (c) a minimal hypersurface of Cartan type, or (d) a product of sphere \(S^1(\sqrt 3/3)\times S^1(\sqrt 3/3) \times S^1(\sqrt 3/3)\).
scientific article; zbMATH DE number 2133429
scientific article