Large dimension behavior of the Hessian eigenvalues of the unit balls

2025/05/14 by Nam Q. Le, Le, Nam Q.
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Sports Dynamics and Biomechanics

paper · pdf · doi:10.48550/arxiv.2505.09409

openalex publication_date 2025/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We show that a sequence of k-Hessian eigenvalues of the unit ball in \mathbb Rⁿ stays bounded as long as the ratio n/k stays bounded. Moreover, we identify their growth of order at least (2-1/k) in n/k. In the case k=n, we show that the Monge--Ampère eigenvalues of the unit balls tend to 4 in the large dimension limit.

Citations

• A Convergent Inexact Abedin-Kitagawa Iteration Method for Monge-Ampère Eigenvalue Problems
• A spectral characterization and an approximation scheme for the Hessian eigenvalue
• Principal eigenvalues for k-Hessian operators by maximum principle methods
• The eigenvalue problem for the Monge-Amp `ere operator on general\n bounded convex domains

Related

• Growth of eigenvalues of Floer Hessians
• Estimates for the higher order buckling eigenvalues in the unit sphere
• Uniform stability of the ball with respect to the first Dirichlet and Neumann ∞-eigenvalues
• Type problem and the first eigenvalue
• Some estimates for the higher eigenvalues of sets close to the ball