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

Yiqian Shi, Bin Xu

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…