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

We initiate a systematic study of four dimensional $\mathcal{N}=2$ 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 $\mathcal{N}=2$ 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 $\mathcal{N}=2$ 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 $\mathcal{N}=2$ supersymmetry-preserving deformations of unitary $\mathcal{N}=2$ 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, 1601.00011, gives the details of the explicit construction of the Coulomb branch geometries discussed here, while the third, 1609.04404, discusses the computation of central charges of the associated SCFTs.