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

Thomas Beck, Spencer Becker-Kahn, Boris Hanin

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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 Rn + 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.

Original languageEnglish (US)
Article number140
JournalCalculus of Variations and Partial Differential Equations
Issue number5
StatePublished - Oct 1 2018
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics


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