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

Abstract

In this paper, we study the principal eigenvalue $\mu ({\mathcal{F}}_k^{-},E)$ of the fully nonlinear operator \[ \mathscr{F}_k^-[u] = \mathcal{P}_k^-(\nabla^2 u) - h |\nabla u| \] on a set $E⋐{\mathbb{R}}^n$, 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 $\mu ({\mathcal{F}}_k^{-},E)$ in terms of a generalized Hausdorff measure ${\mathcal{H}}_{\Psi }(E)$, for suitable $\Psi$ 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.

17 pages. Matemática Contemporânea, volume in honor of Renato Tribuzy for his 75th birthday. Package axessibility included to make the paper available to visually impaired people. Last version: some typos corrected

References in corpus (1)

• Complete proper minimal surfaces in convex bodies of $R^3$