The Bochner Formula via Volume Variations

TL;DR

This paper presents a new derivation of the Bochner formula for the Laplacian using volume variations, offering a pedagogical perspective and discussing potential extensions of the technique.

Contribution

It introduces a volume variation-based derivation of the Bochner formula, providing a novel pedagogical approach and discussing possible extensions.

Findings

The derivation links Laplacian to volume variation along gradient flows.

The approach offers pedagogical value despite limited research impact.

Extensions of the technique are discussed.

Abstract

In this short paper, we re-derive the Bochner formula for the Laplacian by considering local variations of volume. The derivation is rooted in the fact that the Laplacian of a function measures the volume variation along the flow of the gradient vector of the function. Possible extensions of this approach/technique are also discussed. While the value of this approach may be limited in terms of research, we think it definitely has pedagogical value.