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

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 $Σ$ 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 $Σ$ (i.e., $Φ(Σ)$ with $Φ: M \to M$ a diffeomorphism arbitrarily close to the identity in the $C^0$ norm).

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