Estimates for eigensections of Riemannian vector bundles

TL;DR

This paper establishes bounds on the derivatives of Laplace eigensections in Riemannian vector bundles, showing that small eigenvalues correspond to eigensections that are nearly parallel, depending on geometric properties.

Contribution

It provides a new bound relating the covariant derivative of eigensections to geometric quantities, extending understanding of eigensection behavior on Riemannian bundles.

Findings

Bound on the $L^{ abla}$-norm of eigensections' derivatives

Eigensections with small eigenvalues are nearly parallel

Dependence of bounds on diameter, dimension, Ricci curvature, and bundle curvature

Abstract

We derive a bound on the $L^{\infty}$-norm of the covariant derivative of Laplace eigensections on general Riemannian vector bundles depending on the diameter, the dimension, the Ricci curvature of the underlying manifold, and the curvature of the Riemannian vector bundle. Our result implies that eigensections with small eigenvalues are almost parallel.