#### Year of Publication

2014

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Doctoral Dissertation

#### College

Arts and Sciences

#### Department

Mathematics

#### First Advisor

Dr. Peter Hislop

#### Abstract

In 1976, Uhlenbeck used transversality theory to show that for certain families of elliptic operators, the property of having only simple eigenvalues is generic. As one application, she proved that on a closed Riemannian manifold, the eigenvalues of the Laplace-Beltrami operator Δ* _{g}* are all simple for a residual set of

*C*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

^{r}*C*metrics such that the nonzero eigenvalues of the Hodge Laplacian Δ

^{r}*on*

_{g}^{(k)}*k*-forms are all simple for

*0 ≤*

*k*

*≤ 3*. In this dissertation, we continue to address the question of whether Uhlenbeck's theorem can be extended to differential forms. In particular, we prove that for a residual set of

*C*metrics, the nonzero eigenvalues of the Hodge Laplacian Δ

^{r}

_{g}^{(}^{2) }acting on coexact 2-forms on a closed 5-manifold have multiplicity 2. To prove our main result, we structure our argument around a study of the Beltrami operator *

*, which is related to the Hodge Laplacian by Δ*

_{g}d

_{g}^{(}^{2) }= -(*

*)*

_{g}d^{2}when the operators are restricted to coexact 2-forms on a 5-manifold. We use 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*= 4

*ℓ*+ 1 and

*k*= 2

*ℓ*for

*ℓ*

*ϵ*

**N**.

#### Recommended Citation

Gier, Megan E., "EIGENVALUE MULTIPLICITES OF THE HODGE LAPLACIAN ON COEXACT 2-FORMS FOR GENERIC METRICS ON 5-MANIFOLDS" (2014). *Theses and Dissertations--Mathematics*. 14.

https://uknowledge.uky.edu/math_etds/14