\(L^{n/2}\) pinching theorem for submanifolds with parallel mean curvature in \(\mathbb H^{n+p}(-1)\)
The main theorem: Let \(M\) be an \(n\)-dimensional complete submanifold, \(n \geq 3\), with parallel mean curvature in the hyperbolic space \(\mathbb H^{n+p}\). Assume that \(H > 1\) and \(\int_M (S-nH^2)^{n/2}\, dM < C(n,H)\), where \(H\) denotes the mean curvature, \(S\) denotes the squared length of the second fundamental form of \(M\), and \(C(n,H)\) is an (explicitely given) constant. Then \(S = nH^2\), that is \(M\) is the sphere of radius \((H^2 - 1)^{-1/2}\). Using an isometric embedding of \(\mathbb S^{n+p}\) into a horosphere of \(\mathbb H^{n+p+2}\), the following corollary is derived: Let \(M\) be an \(n\)-dimensional complete submanifold, \(n \geq 3\), with parallel mean curvature in the sphere \(\mathbb S^{n+p}\). Assume that \(\int_M (S-nH^2)^{n/2}\, dM < C'(n)\). Then \(M\) is the sphere of radius \((1+H^2)^{-1/2}\).
scientific article; zbMATH DE number 6142349