Nodal sets of smooth functions with finite vanishing order and p-sweepouts

We show that on a compact Riemannian manifold (M, g), nodal sets of linear combinations of any p+ 1 smooth functions form an admissible p-sweepout provided these linear combinations have uniformly bounded vanishing order. This applies in particular to finite linear combinations of Laplace eigenfunctions. As a result, we obtain a new proof of the Gromov, Guth, Marques–Neves upper bounds on the min–max p-widths of M. We also prove that close to a point at which a smooth function on R^{n + 1} vanishes to order k, its nodal set is contained in the union of kW1,p graphs for some p> 1. This implies that the nodal set is locally countably n-rectifiable and has locally finite Hⁿ measure, facts which also follow from a previous result of Bär. Finally, we prove the continuity of the Hausdorff measure of nodal sets under heat flow.