Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary

TL;DR

This paper establishes bounds on the gradient of eigenfunctions of the Laplace-Beltrami operator on compact Riemannian manifolds, showing it scales linearly with the eigenvalue and the eigenfunction's supremum norm.

Contribution

It provides sharp, eigenvalue-dependent bounds on the gradient of eigenfunctions on compact Riemannian manifolds without boundary.

Findings

Gradient of eigenfunctions scales linearly with eigenvalue

Upper and lower bounds depend only on the manifold and eigenvalue

Results are valid for all eigenfunctions with eigenvalue ≥ 1

Abstract

Let $e_\l(x)$ be an eigenfunction with respect to the Laplace-Beltrami operator $\Delta_M$ on a compact Riemannian manifold $M$ without boundary: $\Delta_M e_\l=\l^2 e_\l$. We show the following gradient estimate of $e_\l$: for every $\l\geq 1$, there holds $\l\|e_\l\|_\infty/C\leq \|\nabla e_\l\|_\infty\leq C{\l}\|e_\l\|_\infty$, where $C$ is a positive constant depending only on $M$.