interior of $M$ which near the boundary takes the form $$\label{eq:fake-incomplete-cusp-edge}
  g_{ice} = dx^2 + x^{2k} g_Z + \pi^* g_Y + {\widetilde{g}}, \qquad k > 1,$$ where $g_Y$ is a metric on the base $Y$, $g_Z$ is positive definite restricted to the fibers, $x$ is the distance to the boundary (to first order), and ${\widetilde{g}}$ is a higher order term. Thus near the boundary $(M, g_{ice})$ is a bundle of geometric horns over a smooth Riemannian manifold $Y$. When $k = 3$, such metrics model the singular behavior of the Weil-Petersson metric on the moduli space of Riemann surfaces, as we discuss below.

In this paper, we study the Hodge-Laplacian $$\label{eq:Hodge-Laplacian}
  \Delta := \Delta^{g_{ice}} =  d \delta + \delta d$$ acting on differential forms. Our first result shows that under conditions which contain the main examples of interest, one need not impose ‘ideal boundary conditions’ at ${\partial}M$ in order to obtain a self-adjoint operator.

\[thm:essentiallyselfadjoint\] Let $(M, g_{ice})$ be an incomplete cusp edge manifold that is ‘Witt’, meaning that either $\dim Z = f$ is odd or $$\label{eq:witt}
    H^{f/2}(Z) = \{0\}.$$ Assume furthermore that $g_{ice}$ satisfies below and that the parameter $k$ in satisfies $$\label{eq:k_3}
    k \ge 3.$$ Then the 