Flexibility of Laplace-Beltrami eigenfunctions under metric variation

We study the surjectivity of the differential of the Laplace-Beltrami operator with respect to the metric parameter. We use the obtained results to prove that every small deformation of a nonconstant eigenfunction can be realized with a small perturbation of the metric. As a consequence, we prove that every generic property of smooth functions which is invariant under multiplication by a nonzero scalar is generic for eigenfunctions of the Laplace-Beltrami operator, extending a result by Uhlenbeck.