Let $w_k$ be the eigenfunctions of the Laplace-Beltrami operator on a compact manifold $M$ without boundary. We assume that $\{w_k\}$ are orthonormal, thus $\|w_k \|_{L^2} = 1$.
We know $w_k$ are smooth functions. Is such a bound true: $$\lVert w_k \rVert_{L^\infty(M)} \leq C$$ for all $k$? i.e. are all the eigenfunctions bounded above p/w a.e. by a single constant? Can we remove the a.e. part?
In 1D domains the eigenfunctions are sine and cosine functions which are nice of course.