## Abstract

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 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.

Original language | English (US) |
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Article number | 140 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 57 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2018 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics