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

Author(s)
Beck, Thomas; Becker-Kahn, Spencer; Hanin, Boris
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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.
Date issued
2018-08-28
URI
https://hdl.handle.net/1721.1/131379
Department
Massachusetts Institute of Technology. Department of Mathematics
Publisher
Springer Berlin Heidelberg
Citation
Calculus of Variations and Partial Differential Equations. 2018 Aug 28;57(5):140
Version: Author's final manuscript

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