Let $M$ be an oriented, compact, differentiable manifold with some Riemmanian metric $g$, so that $(M,g)$ has a nice volume form and one can define $L^2(M,g)$ as the completion of $C^\infty(M)$ under the inner product induced by the volume form (one can similarly define $L^2$ spaces of arbitrary forms). Since $M$ is compact, all of the $L^2(M,g)$ are essentially the same for any metric $g$. The Laplacian is defined naturally as $\Delta = -*d*d$, and the Hodge theorem says that its eigenfunctions form a basis for $L^2(M,g)$.

Now if $M = S^1$ and $g$ is the natural metric induced by the covering of $\mathbb{R}$ then $\Delta = -\frac{d}{dx^2}$ and the eigenfunctions are $e^{in\cdot}$. Carleson's theorem states that, unlike an arbitrary basis for $L^2(S^1)$, the convergence of the trigonometric polynomials is pointwise convergence almost everywhere.

Is it suspected that this pointwise almost everywhere convergence of $\Delta$ eigenfunction expansions holds over $L^2(M,g)$ for any compact manifold $(M,g)$ with Laplacian $\Delta$? What about for $L^2$ spaces of forms? Are there any known counter-examples?