Let $(M,g)$ be a closed, smooth Riemannian manifold and suppose that $\phi_1,\dots,\phi_m$ are the first $m$ eigenfunctions of the Laplace-Beltrami operator on $M$. Write $f = \phi_1 + \dots + \phi_m$. How big can the set

$ \mathcal{S} := \left\{ x \in M\ :\ f(x) = 0\ \text{and}\ \nabla f(x) = 0\right\}$ be?

Is it, for example, known that $\mathrm{dim}_{\mathcal{H}}(\mathcal{S}) \leq n-2$?

Let $(M,g)$ be a smooth, closed Riemannian manifold and suppose that $\phi_1,\dots,\phi_m$ are eigenfunctions of the Laplacian on $M$. Write $f = \phi_1 + \dots + \phi_m$.

How big can the set $\mathcal{S} := \left\{ x \in M\ :\ f(x) = \nabla f(x) = 0\right\}$ be? Is it, for example, known that $\mathrm{dim}_{\mathcal{H}}(\mathcal{S}) \leq n-2$?

Some motivation comes from the facts that:

1. The nodal set of $f$ is known to be $(n-1)$-dimensional [Donnelley '94].
2. If the metric is analytic, then for a single eigenfunction $\phi$, the singular set is $(n-2)$-dimensional (and more is known) [Han, Hardt, Lin, M. and T. Hoffmann-Ostenhof, Nadirashvili...].