On the principal eigenvalue of the truncated Laplacian, and submanifolds with bounded mean curvature
Gregório Pacelli F. Bessa, Luquésio Petrola de M. Jorge, Luciano Mari
备注：
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
发表信息：
Mat. Contemp. 49 (2022), Special issue in honor of Professor Renato de Azevedo Tribuzy on the occasion of his 75th birthday, 212-235
本文研究集合$E \Subset \mathbb{R}^n$上完全非线性算子$\mathscr{F}_k^-[u] = \mathcal{P}_k^-(\nabla^2 u) - h |\nabla u|$的主特征值$μ(\mathscr{F}_k^-,E)$，其中$h \in [0,\infty)$，$\mathcal{P}_k^-(\nabla^2 u)$是黑塞矩阵$\nabla^2 u$的最小$k$个特征值之和。我们借助广义豪斯多夫测度$\mathscr{H}_\Psi(E)$（$\Psi$是依赖于$k$的合适函数）证明了$μ(\mathscr{F}_k^-,E)$的一个下界估计，向猜想中的精确估计$μ(\mathscr{F}_k^-,E) \ge C \mathscr{H}^k(E)^{-2/k}$推进了若干步。该定理被用于研究$\mathbb{R}^n$中有界子流形的谱，针对丘成桐提出的一个问题，改进了我们此前的工作。特别地，该结果适用于常平均曲率（CMC）曲面的普拉托（Plateau）问题的解。
英文摘要：
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.
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