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

arXiv:1503.05105v1We 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,g0) of dimension d≥3, there is a metric g on M conformally equivalent to g0 and with the same volume such that the nodal set of its first nontrivial eigenfunction is a C0 -small deformation of Σ (i.e., Φ(Σ) with Φ:M→M a diffeomorphism arbitrarily close to the identity in the C0 norm).This work was supported by the ERC Starting Grants [633152 to A.E. and 335079 to D.P.-S.] and in part by the ICMAT–Severo Ochoa grant [SEV-2015-0554 to A.E. and D.P.-S.] and was partially supported by CRC1060 of the DFG (to S.S.).Peer Reviewe