Nodal sets of smooth functions with finite vanishing order and p-sweepouts
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Author(s)
Beck, Thomas
Becker-Kahn, Spencer
Hanin, Boris
Date Issued
August 28, 2018
Publisher
Springer Berlin Heidelberg
Citation
Calculus of Variations and Partial Differential Equations. 2018 Aug 28;57(5):140
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00526-018-1406-y
