TY - JOUR

T1 - Nodal sets of smooth functions with finite vanishing order and p-sweepouts

AU - Beck, Thomas

AU - Becker-Kahn, Spencer

AU - Hanin, Boris

N1 - Funding Information:
BH is partially supported by NSF Grant DMS-1400822.
Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s00526-018-1406-y

DO - 10.1007/s00526-018-1406-y

M3 - Article

AN - SCOPUS:85052601965

SN - 0944-2669

VL - 57

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

IS - 5

M1 - 140

ER -