Let \((M,g)\) be a compact smooth Riemannian \(d\)-manifold with \(\Ric(M)\ge 0\) and \(\nabla\) the corresponding Levi-Civita connection . Furthermore, let \(u_1,\ldots ,u_n\in C^\infty(M)\) be eigenfunctions of the Laplacian with the respective eigenvalues \(\lambda_1,\ldots ,\lambda_n\). We denote the maximal eigenvalue with \(E\). Assume there exists a Bernstein inequality on \(M\). Then, for every \(u=\sum_{i=1}^{n} a_i u_i\) with \(a_i\in \mathbb{R}\), there exists a finite sequence of geodesic balls \(B_1,\ldots ,B_N\) with radius \(r>0\) and a constant \(C>0\) which depends on \(M\), such that a local Bernstein inequality holds:
\begin{equation*} \lVert \nabla^k u\rVert_{L^2(B_i)}^2\le (CE)^k \lVert u\rVert_{L^2(B_i)}^2, \end{equation*}for every \(k\ge 1\) and \(i \in \{1,\ldots ,N\}\) and
\begin{equation*} \lVert u\rVert_{L^2(\bigcup_{i=1}^N B_i)}^2 \ge \frac{1}{2} \lVert u\rVert_{L^2(M)}^2. \end{equation*}We call such balls good balls.
According to (0x6762cf17) there is a finite cover \((B_i)_{i\in I}\) of \(M\) consisting of geodesic balls with radius \(r\) and the covering multiplicity \(\kappa\) is bounded by some constant \(C=C(M)>0\).
We call balls, which are not good, bad. To be more precise for every bad ball \(B\), there exists a \(k>1\) such that
\begin{equation*} \lVert \nabla^k u\rVert_{L^2(B)}^2\ge (CE)^k \lVert u\rVert_{L^2(B)}^2. \end{equation*}Note, we specify \(C>0\) later on.
Then
\begin{align*} \int_{\bigcup_{B \text{ is bad }}} u^2 &\le \int_{\bigcup_{B \text{ is bad }}} \sum_{k=1}^{\infty} \frac{1}{C^{k}E^{k}} \langle \nabla^k u, \nabla^k u\rangle \\ &\le \sum_{k=1}^{\infty} \frac{1}{C^{k}E^{k}} \int_{\bigcup_{B \text{ is bad }}} \langle \nabla^k u, \nabla^k u\rangle \\ &\le \sum_{k=1}^{\infty} \frac{\kappa}{C^{k}E^{k}} \int_{M} \langle \nabla^k u, \nabla^k u\rangle \\ &\le \sum_{k=1}^{\infty} \frac{\kappa}{C^{k}} \int_{M} u^2 \\ &\le \frac{\kappa}{C-1} \int_{M} u^2, \end{align*}where we used the Bernstein inequality .
Choose \(C=2\kappa +1\) and we obtain
\begin{equation*} \int_{\bigcup_{B \text{ is bad }}} u^2 \le \frac{1}{2} \int_{M} u^2. \end{equation*}