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

In this paper, we study the principal eigenvalue $\mu ({\mathcal{F}}_k^{-},E)$ of the fully nonlinear operator $${\mathcal{F}}_k^{-}[u]={\mathcal{P}}_k^{-}({\mathrm{\nabla }}^{2}u)-h\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }$$ on a set $E⋐{\mathbb{R}}^n$, where $h\in [0,\mathrm{\infty })$ and ${\mathcal{P}}_k^{-}({\mathrm{\nabla }}^{2}u)$ is the sum of the smallest $k$ eigenvalues of the Hessian ${\mathrm{\nabla }}^{2}u$. We prove a lower estimate for $\mu ({\mathcal{F}}_k^{-},E)$ in terms of a generalized Hausdorff measure ${\mathcal{H}}_{\mathrm{\Psi }}(E)$, for suitable $\mathrm{\Psi }$ depending on $k$, moving some steps in the direction of the conjecturally sharp estimate $$\mu ({\mathcal{F}}_k^{-},E)\ge C{\mathcal{H}}^k(E{)}^{-2\mathrm{/}k}\mathrm{.}$$ 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.