On the second eigenvalue of the Laplace operator penalized by curvature

Consider the operator -del(2) - q(kappa), where -del(2) is the (positive) Laplace-Beltrami operator on a closed manifold of the topological type of the two-sphere S-2 and q is a symmetric non-negative quadratic form in the principal curvatures. Generalizing a well-known theorem of J. Hersch for the Laplace-Beltrami operator alone, it is shown in this note that the second eigenvalue lambda(1) is uniquely maximized, among manifolds of fixed area, by the true sphere.