Generalization of Hadamardʼs Laplace Eigenvalue Formula to Deforming Manifolds

Abstract

For a Euclidean domain with a moving boundary, Hadamardʼs formula relates the rate of change of the Laplace eigenvalues to the normal velocity of the boundary. We generalize Hadamardʼs formula to deforming Riemannian maniforlds with contour boundary moving in a compatible manner. Our analysis finds direct applications in the dynamics of fluid films. The spectrum of the surface Laplacian describes the frequencies of normal oscillations of the filmʼs surface as well as tangential oscillations in thickness.