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

2021/09/29 by G. Pacelli Bessa, Bessa, Gregório Pacelli F., Luquésio P. Jorge +3
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2109.14740

openalex publication_date 2021/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the principal eigenvalue μ(\mathscrF_k^-,E) of the fully nonlinear operator \mathscrF_k^-[u] = P_k^-(∇² u) - h |∇ u| on a set E \Subset ℝⁿ, where h ∈ [0,∞) and P_k^-(∇² u) is the sum of the smallest k eigenvalues of the Hessian ∇² u. We prove a lower estimate for μ(\mathscrF_k^-,E) in terms of a generalized Hausdorff measure \mathscrH_Ψ(E), for suitable Ψ depending on k, moving some steps in the direction of the conjecturally sharp estimate μ(\mathscrF_k^-,E) ≥ C \mathscrH^k(E)^-2/k. The theorem is used to study the spectrum of bounded submanifolds in ℝⁿ, 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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