Title:  Eigenvalue Multiplicities of the Hodge Laplacian on Coexact 2-Forms for Generic Metrics on 5-Manifolds

Abstract:  In 1976, Uhlenbeck used transversality theory to show that on a closed Riemannian manifold, the eigenvalues of the Laplace-Beltrami operator are all simple for a residual set of C^r metrics. In 2012, Enciso and Peralta-Salas established an analogue of Uhlenbeck's theorem for differential forms, showing that on a closed 3-manifold, there exists a residual set of C^r metrics such that the nonzero eigenvalues of the Hodge Laplacian on k forms are all simple.  We continue to address the question of whether Uhlenbeck's theorem can be extended to differential forms by proving that for a residual set of C^r metrics, the nonzero eigenvalues of the Hodge Laplacian acting on coexact 2-forms on a closed 5-manifold have multiplicity 2.  We structure our argument around a study of the Beltrami operator, using techniques from perturbation theory to show that the Beltrami operator has only simple eigenvalues for a residual set of metrics.  We further establish even eigenvalue multiplicities for the Hodge Laplacian acting on coexact k-forms in the more general setting n=4m+1 and k=2m.