ECH capacities of concave singular toric domains

By definition, a toric domain has a boundary contact manifold diffeomorphic to a three dimensional sphere. In the present work we extend the definition of the toric domains in dimension four so that it admits a contact manifold diffeomorphic to a lens space L(n, 1). We call them 'singular toric domains' since they naturally have an orbifold point. We calculate the ECH capacities for a specific subfamily of these singular toric domains that generalize the concave toric domains. Interestingly, even though we are calculating the capacities of an orbifold, the result can be adapted to study embedding problems in some of the desingularizations of these spaces, for example the unitary cotangent bundle of two dimensional sphere or the unitary cotangent bundle of projective plane.