On the principal eigenvalue of the truncated Laplacian, and submanifolds with bounded mean curvature
In this paper, we study the principal eigenvalue $μ(\mathscr{F}_k^-,E)$ 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 $μ(\mathscr{F}_k^-,E)$ in terms of a generalized Hausdorff measure $\mathscr{H}_Ψ(E)$, for suitable $Ψ$ depending on $k$, moving some steps in the direction of the conjecturally sharp estimate \[ μ(\mathscr{F}_k^-,E) \ge C \mathscr{H}^k(E)^{-2/k}. \] 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.