Upper and lower bounds of the Hausdorff measure of nodal sets of Laplace eigenfunctions have been largely studied in the context of smooth Riemannian manifolds, while very little is known in the context of possibly singular spaces. We investigate this problem following a technique introduced by Steinerberger, based on indeterminacy estimates and bounds on the Wasserstein distance between the positive and negative parts of an eigenfunction. This approach allows us to obtain results also in the setting of metric measure spaces satisfying a notion of Ricci curvature bounded below.

This is based on joint works with Fabio Cavalletti and Nicolò De Ponti.