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arXiv · 1503.05105

Prescribing the nodal set of the first eigenfunction in each conformal class

Alberto Enciso·Daniel Peralta-Salas·Stefan Steinerberger
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Abstract

We consider the problem of prescribing the nodal set of the first nontrivial eigenfunction of the Laplacian in a conformal class. Our main result is that, given a separating closed hypersurface $\Sigma$ in a compact Riemannian manifold $(M,g_0)$ of dimension $d \geq 3$, there is a metric $g$ on $M$ conformally equivalent to $g_0$ and with the same volume such that the nodal set of its first nontrivial eigenfunction is a $C^0$-small deformation of $\Sigma$ (i.e., $\Phi(\Sigma)$ with $\Phi : M \to M$ a diffeomorphism arbitrarily close to the identity in the $C^0$ norm).

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Alberto Enciso, Daniel Peralta-Salas, Stefan Steinerberger. 2015-03-17. Prescribing the nodal set of the first eigenfunction in each conformal class. https://arxiv.org/abs/1503.05105

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