A BLW inequality for vector bundles and applications to spectral bounds
The authors prove a non-commutative Bochner-Lichnerowicz-Weitzenböck formula for the Bochner Laplacian \(B\) acting on sections of a real finite dimensional \(C^*\)-algebra bundle. Then they deduce a lower bound for the spectrum \(\sigma(-B)\) of \(-B\) acting on a Euclidean vector bundle which has no non-vanishing sections. If \(E_{\mathrm{sc}}\) denotes the smallest non-zero eigenvalue of the scalar Laplacian then for \(E\in \sigma(-B)\) either \(E>1/d^2\) or \(E\geq c\cdot E_{\mathrm{sc}}\) where \(d=\mathrm{diam } M\) and \(c\) is an explicit constant depending on the Ricci curvature of the manifold \(M\) and on the curvature and the current of the bundle.
Bochner-Lichnerowicz-Weitzenböck formula for the Bochner Laplacian
scientific article; zbMATH DE number 4140892
scientific article