On the principal eigenvalue of the truncated Laplacian, and submanifolds with bounded mean curvature

In this paper, we study the principal eigenvalue $mu(FF_k^-,E)$ of the fully nonlinear operator [ F_k^-[u] = P_k^-( abla^2 u) - h | abla u| ] on a set $E Subset mathbb{R}^n$, where $h in [0,infty)$ and $P_k^-( abla^2 u)$ is the sum of the smallest $k$ eigenvalues of the Hessian $ abla^2 u$. We prove a lower estimate for $mu(FF_k^-,E)$ in terms of a generalized Hausdorff measure $mathcal{H}_Psi(E)$, for suitable $Psi$ depending on $k$, moving some steps towards the conjecturally sharp estimate [ mu(F_k^-,E) ge C mathcal{H}^k(E)^{-2/k}. ] The theorem is used to study the spectrum of bounded submanifolds in $R^n$, 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.