On the nodal set of solutions to Dirac equations

Abstract

Motivated by various geometric problems, we study the nodal set of solutions to Dirac equations on manifolds, of general form. We prove that such set has Hausdorff dimension less than or equal to $n-2$, $n$ being the ambient dimension. We extend this result, previously known only in the smooth case or in specific cases, working with locally Lipschitz coefficients. Under some additional, but still quite general, structural assumptions we provide a stratification result for the nodal set, which appears to be new already in the smooth case. This is achieved by exploiting the properties of a suitable Almgren-type frequency function, which is of independent interest.