  In this paper, we study the principal eigenvalue $\mu(\mathscr{F}_k^-,E)$ of
the fully nonlinear operator
  \[
  \mathscr{F}_k^-[u] = \mathcal{P}_k^-(\nabla^2 u) - h |\nabla u|
  \] on a set $E \Subset \mathbb{R}^n$, where $h \in [0,\infty)$ and
$\mathcal{P}_k^-(\nabla^2 u)$ is the sum of the smallest $k$ eigenvalues of the
Hessian $\nabla^2 u$. We prove a lower estimate for $\mu(\mathscr{F}_k^-,E)$ in
terms of a generalized Hausdorff measure $\mathscr{H}_\Psi(E)$, for suitable
$\Psi$ depending on $k$, moving some steps in the direction of the
conjecturally sharp estimate
  \[
  \mu(\mathscr{F}_k^-,E) \ge C \mathscr{H}^k(E)^{-2/k}.
  \] The theorem is used to study the spectrum of bounded submanifolds in
$\mathbb{R}^n$, improving on our previous work in the direction of a question
posed by S.T. Yau. In particular, the result applies to solutions of Plateau's
problem for CMC surfaces.
