Generalized Clifford torus in \(S^{n+1}\) and prescribed mean curvature function (Q808006)

An affirmative answer is presented to the open problem: ``Let \(H\) be a function on \(S^{n+1}\) \((n>2)\). Find conditions on \(H\) to insure that there exists a closed hypersurface in \(S^{n+1}\) which is homeomorphic to the Clifford torus \(S^m\times S^{n-m}\) \((2\leq m\leq n-1)\) and has the mean curvature equal to \(H\)''. Now, set \(N=\{X=Sr\in\mathbb R^{m+1}\mid 0<S<1\},\) where \(r\) is the position vector of \(S^m(1)<\mathbb R^{m+1}\) \((2\leq m\leq n-1),\) and put the following two conditions on \(H\): (1) There are constants \(S_1\), \(S_2\), where \(1>S_1\geq \sqrt{1/2}\geq S_2>0,\) such that \(H(Sr)<(m-nS^2)/nS\sqrt{1-S^2}\) when \(1>S>S_1\), and \(H(Sr)>(m-nS^2)/ns\sqrt{1-S^2}\) when \(S_2>S>0\). (2) Set \(P=\{Sr\in N\mid S_2\leq S\leq S_1\}\), \((\partial /\partial S)[SH(Sr)]\leq 0\), in \(P\). It is then shown that there is a hypersurface \(M_1\times M_2\) in \(S^{n+1}\) whose mean curvature is given precisely by \(H(Sr)\) and depends only on \(M_1\), where \(M_1\) is homeomorphic to \(S^m(1)\), \(M_2\) is homeomorphic to \(S^{n-m}(1)\).