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

Yiqian Shi; Bin Xu

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arXiv · 0905.1366

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

Yiqian Shi·Bin Xu

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Abstract

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

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BibTeX RIS

Yiqian Shi, Bin Xu. 2009-05-21. Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary. https://arxiv.org/abs/0905.1366

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