A global pinching theorem for surfaces with constant mean curvature in \(S^3\) (Q2750899)

Let \(M\) be a compact immersed hypersurface in the unit sphere \(S^{n+1}\) with constant mean curvature \(H\) and let \(\Phi\) denote the square of the length of the tensor \(\varphi_{ij}= h_{ij}- (H/n)\delta_{ij}\), where \(h= (h_{ij})\) is the second fundamental form of \(M\).NEWLINENEWLINENEWLINEThe main result of this paper is the following: Let \(M\) be a compact immersed surface in the unit sphere \(S^3\) with constant mean curvature \(H\). Then NEWLINE\[NEWLINE\|\Phi \|_2\geq 2\pi\sqrt {2g/m(B)}NEWLINE\]NEWLINE where \(g\) is the genus of \(M\), \(\|\cdot \|_2\) is the \(L^2\)-norm and \(m(B)\) is a function of \(B=2+ (1/2)H^2\) introduced by the authors. The above theorem is connected with the first author's result in [Proc. Am. Math. Soc. 113, 1041-1044 (1991; Zbl 0739.53046)].