Small eigenvalues of the Hodge-Laplacian with sectional curvature bounded below

2025/06/13 by Anné, Colette, Takahashi, Junya · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.11579

Abstract

For each degree p and each natural number k ≥ 1, we construct a oneparameter family of Riemannian metrics on any oriented closed manifold with volume one and the sectional curvature bounded below such that the k-th positive eigenvalue of the Hodge-Laplacian acting on differential p-forms converges to zero. This result imposes a constraint on the sectional curvature for our previous result in [AT24].

Citations

• Gap phenomena under curvature restrictions
• Poincaré inequality for one forms on four manifolds with bounded Ricci curvature
• Small eigenvalues of the rough and Hodge Laplacians under fixed volume
• A Cheeger-like inequality for coexact 1-forms
• Eigenvalue estimates and differential form Laplacians on Alexandrov\n spaces
• Spectral convergence under bounded Ricci curvature
• Perelman's Stability Theorem
• Discretization of Riemannian manifolds applied to the Hodge Laplacian
• Remark about the spectrum of the p-form Laplacian under a collapse with curvature bounded below

Cited by

• Eigenvalue Estimate for the Rough Laplacian on 1-Forms and its Applications
• A note on the Maxwell's eigenvalues on thin sets

Related

• Small eigenvalues of the rough and Hodge Laplacians under fixed volume
• On the first eigenvalue of the Hodge Laplacian of submanifolds
• Gradient estimate for eigenforms of Hodge Laplacian
• Lower bounds for the eigenvalues of the Hodge Laplacian on certain non-convex domains
• Discretization of Riemannian manifolds applied to the Hodge Laplacian