Prescribing the Nodal Set of the First Eigenfunction in Each Conformal Class

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).