Geometric constraints on the space of $N=2$ SCFTs. Part I: physical constraints on relevant deformations

We initiate a systematic study of four dimensional $N=2$ superconformal field theories (SCFTs) based on the analysis of their Coulomb branch geometries. Because these SCFTs are not uniquely characterized by their scale-invariant Coulomb branch geometries we also need information on their deformations. We construct all inequivalent such deformations preserving $N=2$ supersymmetry and additional physical consistency conditions in the rank 1 case. These not only include all the ones previously predicted by S-duality, but also 16 additional deformations satisfying all the known $N=2$ low energy consistency conditions. All but two of these additonal deformations have recently been identified with new rank 1 SCFTs; these identifications are briefly reviewed. Some novel ingredients which are important for this study include: a discussion of RG-flows in the presence of a moduli space of vacua; a classification of local $N=2$ supersymmetry-preserving deformations of unitary $N=2$ SCFTs; and an analysis of charge normalizations and the Dirac quantization condition on Coulomb branches. This paper is the first in a series of three. The second paper https://doi.org/10.1007/JHEP02(2018)002 gives the details of the explicit construction of the Coulomb branch geometries discussed here, while the third https://doi.org/10.1007/JHEP02(2018)003 discusses the computation of central charges of the associated SCFTs.