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

Abstract

Abstract We show that on a compact Riemannian manifold (M, g), nodal sets of linear combinations of any

$$p+1$$

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

$$\mathbb {R}^{n+1}$$

R

n + 1

vanishes to order k, its nodal set is contained in the union of

$$k\,W^{1,p}$$

k

W

1 , p

graphs for some

$$p > 1$$

p > 1

. This implies that the nodal set is locally countably n-rectifiable and has locally finite

$$\mathcal {H}^n$$

H

n

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.