On a Laplacian which acts on symmetric tensors

TL;DR

This paper explores a Laplacian operator acting on symmetric tensors, analyzing its properties, Weitzenb"ock decomposition, and applications to vanishing theorems and eigenvalue estimates, extending classical differential geometry tools.

Contribution

It introduces and studies a Laplacian on symmetric tensors, analogous to the Hodge-de Rham Laplacian, including its Weitzenb"ock decomposition and analytical properties.

Findings

The Laplacian admits a Weitzenb"ock decomposition.

Vanishing theorems are proved for the null space of the operator.

Lower bounds for the lowest eigenvalue are established.

Abstract

In the present paper we show properties of a little-known Laplacian operator acting on symmetric tensors. This operator is an analogue of the well known Hodge-de Rham Laplacian which acts on exterior differential forms. Moreover, this operator admits the Weitzenb\"ock decomposition and we study it using the analytical method, due to Bochner, of proving vanishing theorems for the null space of a Laplace operator admitting a Weitzenb\"ock decomposition and further of estimating its lowest eigenvalue.