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

• Main Authors: Beck, Thomas (Author), Becker-Kahn, Spencer (Author), Hanin, Boris (Author)
• Format: Article
• Language: English
• Published: Springer Berlin Heidelberg, 2021-09-20T17:16:50Z.
• Subjects: Article
• Online Access: Get fulltext

 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.