Submanifolds in \(S^{n+p}\) with parallel Möbius form
Let \(x: M\to S^{n+p}\) (\(n\geq 2\), \(p\geq 1\)) be an \(n\)-dimensional submanifold in the unit sphere \(S^{n+p}\) without umbilic points and with parallel Möbius form \(\Phi\); let the normalized scalar curvature \(\kappa\) of the Möbius invariant metric \(G\) be constant, and denote by \(\widetilde A\) the traceless Blaschke form. Then the following 3 rigidity theorems are valid: Theorem 1.1: Assume, additionally: \(p\geq 2\), \(\|\widetilde A\|\leq n^{1/2}(n- 1)^{1/2}(n- 2)^{-1}(\kappa- f(n))\) with \(f(n)= (3n- 5)/(2n^2)\), and \(M\) be compact. Then \(M\) is Möbius equivalent to the Veronese surface in \(S^4\). Theorem 1.2: Assume, additionally: \(n\geq 3\), \(p=1\), and \(\|\widetilde A\|\leq n^{1/2}(n-1)^{1/2}(n- 2)^{-1}(\kappa- f(n))\) with \(f(n)= (n- 2)/(n^2)\). Then \(M\) is Möbius equivalent either to the Clifford minimal hypersurface in \(S^{n+1}\) (case: \(\kappa= f(n)\)) or to a torus \(S^1(a)\times S^{n-1}(b)\to S^{n+1}\) (case: \(f(n)< \kappa< n\cdot f(n)\)) with certain radii \(a= a(n,\kappa)\), \(b= b(n,\kappa)\). Theorem 1.3: Assume, additionally: \(n=2\), \(p=1\), \(\kappa\geq 0\) (and \(\kappa=\text{const}\)), and let \(M\) be compact. Then \(M\) is Möbius equivalent to a flat torus in \(S^3\to\mathbb{R}^4\).
scientific article; zbMATH DE number 2151012