Isoperiodic foliation of the stratum $\mathrm{\Omega }{\mathcal{M}}_1(1,1,-2)$

This paper describes the geometry and topology of leaves of the isoperiodic foliation of the stratum $\mathrm{\Omega }{\mathcal{M}}_1(1,1,-2)$. We prove that each leaf is a surface of infinite genus homeomorphic to the Loch Ness monster surface and supports a singular Euclidean structure whose singularities correspond to isoperiodic forms in the lower stratum $\mathrm{\Omega }{\mathcal{M}}_1(2,-2)$. Along the way we also characterize the possible groups arising as the Veech group of a leaf and give a description of the large-scale conformal geometry of the wall-and-chamber decomposition of the leaves.