On the PPW Conjecture For Hopf-symmetric Sets In Non-compact Rank One Symmetric Space

2025/05/24 by Xia, Yusen
#53 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.18439

Abstract

In this paper, we proved that for a bounded Hopf-symmetric domain Ω in a noncompact rank one symmetric space M, the second Dirichlet eigenvalue λ₂ (Ω) ≤ λ₂ (B₁) where B₁ is a geodesic ball in M such that λ₁ (Ω) =λ₁ (B₁). This generalizes the work of Ashbaugh & Benguria, Benguria & Linde for bounded domains in constant curvature spaces.

Citations

• Approaching the isoperimetric problem in H^m_ℂ via the hyperbolic log-convex density conjecture
• The fundamental gap of horoconvex domains in \mathbb Hⁿ
• Isoperimetry and volume preserving stability in real projective spaces
• A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces
• A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space
• A Sharp Bound for the Ratio of the First Two Dirichlet Eigenvalues of a Domain in a Hemisphere of Sⁿ

Related

• A Spectral Characterization Of Geodesic Balls In Non-Compact Rank One Symmetric Spaces
• The second Robin eigenvalue in non-compact rank-1 symmetric spaces
• Isoperimetric inequalities for Neumann eigenvalues on bounded domains in rank-1 symmetric spaces
• Sharp upper bound and a comparison theorem for the first nonzero Steklov eigenvalue
• An isoperimetric inequality for the harmonic mean of Steklov eigenvalues in rank one symmetric spaces