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

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

2009-05-09

arxiv.org/abs/0905.1366v2

Mathematics

Spectral Theory

8 pages. The abstract is shortened to two sentences. The reference of the book by Yu Safarov and D. Vassiliev was added. An al

Scientific paper

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 $M$.

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