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

TL;DR

This paper investigates the principal eigenvalue of a nonlinear operator related to the Laplacian, providing lower bounds connected to geometric measures, and applies these results to the study of submanifolds with bounded mean curvature, including CMC surfaces.

Contribution

It establishes a lower estimate for the principal eigenvalue of a nonlinear operator in terms of a generalized Hausdorff measure, advancing understanding of spectral properties of submanifolds with bounded mean curvature.

Findings

Derived a lower bound for the eigenvalue using generalized Hausdorff measures.

Connected spectral estimates to geometric properties of submanifolds.

Applied results to Plateau's problem for constant mean curvature surfaces.

Abstract

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.