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

TL;DR

This paper demonstrates that for any separating hypersurface in a compact manifold, one can find a conformal metric with the same volume whose first eigenfunction's nodal set closely approximates that hypersurface.

Contribution

It establishes the existence of conformal metrics with prescribed nodal sets near any given separating hypersurface in higher-dimensional manifolds.

Findings

Existence of conformal metrics with prescribed nodal sets

Nodal sets can be made arbitrarily close to a given hypersurface

Results apply to manifolds of dimension at least 3

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).