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

Abstract

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

8 pages. The abstract is shortened to two sentences. The reference of the book by Yu Safarov and D. Vassiliev was added. An alternative proof of the gradient estimate for the unit band spectral projection operator is added in Section 4. The layout is changed