The properties of nodal sets (i.e. zero level sets of eigenfunctions) for the first non-trivial eigenfunction for Laplacians have been studied extensively.
My rough understanding is that one could pick/find/construct a metric for which the nodal sets are what one desires, under some assumptions.
I am looking for some papers where this is explicitly done, via analytical or numerical methods.
Nodal sets usually refer to the zero sets of eigenfunctions - not just arbitrary level sets as mentioned above.
The problem of prescribing the nodal set has been investigated intensely. You might want to have a look at, for example, the papers by Enciso and Peralta-Salas (arXiv version: https://arxiv.org/pdf/1404.1039v1.pdf); Enciso, Peralta-Salas and Steinerberger (arXiv version: https://arxiv.org/pdf/1503.05105v1.pdf); Bourgain and Rudnick (arXiv version: https://arxiv.org/abs/1003.1743); etc.
Needless to say, most of the references in the papers above lead to further results that may also be helpful.
