The Lichnerowicz-Type Laplacians: Vanishing Theorems for Their Kernels and Estimate Theorems for Their Smallest Eigenvalues

Abstract

In the present paper, we prove several vanishing theorems for the kernel of the Lichnerowicz-type Laplacian and provide estimates for its lowest eigenvalue on closed Riemannian manifolds. As an example of the Lichnerowicz-type Laplacian, we consider the Hodge–de Rham Laplacian acting on forms and ordinary Lichnerowicz Laplacian acting on symmetric tensors. Additionally, we prove vanishing theorems for the null spaces of these Laplacians and find estimates for their lowest eigenvalues on closed Riemannian manifolds with suitably bounded curvature operators of the first kind, sectional and Ricci curvatures. Specifically, we will prove our version of the famous differential sphere theorem, which we will apply to the aforementioned problems concerning the ordinary Lichnerowicz Laplacian.

Keywords

• Lichnerowicz Laplacian
• symmetric tensors
• exterior differential form
• vanishing theorem
• eigenvalue