Lowest eigenvalues and formally self-adjoint fourth order elliptic differential operators

Let $(M,g)$ be a closed, smooth, Riemannian manifold of dimension $m\ge 1$. Let $\eta$ be a smooth $(0,1)$-tensor field on $M$. The divergence of $\eta$ is defined as ${\text{div}}_g(\eta ):=g^{ij}(\mathrm{\nabla }\eta {)}_{ij}$. Now let ${\mathrm{\Delta }}_g$ be a differential operator on $M$ that is given on functions by ${\mathrm{\Delta }}_{g}u={\text{div}}_g\mathrm{\nabla }u$. We will call ${\mathrm{\Delta }}_g$ the Laplace-Beltrami operator. With this definition in place, it is not difficult to produce an example of a formally self-adjoint elliptic differential operator on $M$ that has a sign-changing eigenfunction that is associated with the operator’s lowest eigenvalue. Indeed, let ${\lambda }_2$ be the second lowest eigenvalue of $-{\mathrm{\Delta }}_g$, and let $L_g$ be a differential operator on $M$ that is given on functions by $L_{g}u={\mathrm{\Delta }}_g^{2}u+{\lambda }_2\mathrm{\Delta }u$. Then $L_g$ will possess a sign-changing eigenfunction that is associated with $L_g$’s lowest eigenvalue… The question that remains is given a smooth, closed manifold $M$ of dimension $m\ge 1$, how rare are formally self-adjoint elliptic differential operators on $M$ that have sign-changing eigenfunctions that are associated with the operators’ lowest eigenvalues. In this paper, we will see that if $A=T-\lambda g$, where $T$ is a smooth, symmetric, negative semi-definite $(0,2)$-tensor field on $M$, then $P_g$, the differential operator on $M$ given on functions by $P_{g}u=,{\mathrm{\Delta }}_g^2-{\text{div}}_g(A(\mathrm{\nabla }u{)}^{\mathrm{♯}})$, will have the property that it possesses a sign-changing eigenfunction that is associated with the lowest eigenvalue of the operator. This suggests that on any smooth, closed manifold of dimension $m\ge 1$ there exists a lot of formally self-adjoint fourth order elliptic differential operators on the manifold that possess sign-changing eigenfunctions that are associated with the lowest eigenvalues of the operators.