Mathematics

Zero sets of eigenfunctions of the Laplace operator, called nodal sets, have been largely studied in the context of smooth $n$-dimensional Riemannian manifolds. A particular attention has been given to upper and lower bounds for the $(n-1)$-dimensional Hausdorff measure of nodal sets. We investigated this problem in the context of singular spaces satisfying syntethic curvature conditions like $\mathsf{CD}(K,N)$  or $\mathsf{MCP}(K,N)$. In particular we proved a lower bound for the measure of the nodal set. The argument runs through an indeterminacy inequality which involves optimal trasport, following a recent approach due to Steinerberger. 

This is a joint work with Prof. Fabio Cavalletti.