Lemma 4.14 (Uniform estimate for eigenfunctions) . [03HU]

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Lemma 4.14 (Uniform estimate for eigenfunctions).

Let $\{\varphi_{k}\}_{k=1}^{\infty}$ be the eigenfunctions of $\Delta_{h_{0}}$ on $(Y^{3},h_{0})$ with $\|\varphi_{k}\|_{L^{2}(Y^{3})}=1$ , then there exists $C>0$ which depends only on the metric $h_{0}$ such that

(4.146)

\|\varphi_{k}\|_{L^{\infty}(Y^{3})}\leq C\cdot\Lambda_{k}.