Relationship between sectional curvature and null spaces of Lichnerowicz-type Laplacians and their smallest eigenvalues

2023/08/15 by С. Е. Степанов, Stepanov, S. E.
Mathematics · Medicine · #53C20 #Advanced Neuroimaging Techniques and Applications #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2308.07894

openalex publication_date 2023/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The first variant of this article contained a fatal error. Therefore, we publish second version our paper. In the present paper, we prove that the curvature operator of the second kind of a Riemannian manifold is strictly positive if its sectional curvature is strictly positive and the Ricci curvature suitably pinched. In addition, we prove several vanishing theorems for null spaces of the Lichnerowicz, Sampson, and Hodge-de Rham Laplacians and find estimates for their lowest eigenvalues on compact (without boundary) Riemannian manifolds with sectional pinched curvature.

Related

• Small eigenvalues of the Hodge-Laplacian with sectional curvature bounded below
• Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature
• Manifolds with bounded integral curvature and no positive eigenvalue lower bounds
• Small eigenvalues of the rough and Hodge Laplacians under fixed volume
• Geometry in the large of the kernel of Lichnerowicz Laplacians and its applications