A characterization of Clifford tori with constant scalar curvature one by the first stability eigenvalue
An immersion of an \(n\)-dimensional manifold M into the \((n+1)\)-dimensional sphere with constant scalar curvature one, under the metric induced by the sphere, is considered. \(M\) is taken to be a closed orientable manifold. Such surfaces can be characterized variationally as critical points of the integral of the mean curvature \(H\) over \(M\). A sharp upper bound for the first eigenvalue of the corresponding Jacobi operator is given in terms of \(H\), and it is shown that the upper bound is attained only by the Clifford tori in the \((n+1)\)-dimensional sphere. It is noted that the Jacobi operator is elliptic if and only if \(n\) is at least 3 and the third elementary symmetric function of the principal curvatures does not vanish, thus these two conditions are assumed. It then follows that \(H\) can be taken to be positive. Once the Jacobi operator is established to be elliptic, a min-max characterization of its first eigenvalue, using the Rayleigh quotient, can be applied. The primary result of the paper is that the first eigenvalue is less than or equal to \(-2n(n-1)\) times the minimum of \(H\), with equality holding only for the Clifford tori. This work is motived by similar results for minimal hypersurfaces in spheres by Simons, Chern do Carmo, Kobayashi, Lawson, Perdomo, Wu and Santos. The authors dedicate this paper to the memory of Jose F. Escobar.
scientific article; zbMATH DE number 2133667