  A classical aspect of Riemannian geometry is the study of estimates that hold
uniformly over some class of metrics. The best known examples are eigenvalue
bounds under curvature assumptions. In this paper, we study the family of all
left-invariant geometries on $\mathrm{SU}(2)$. We show that left-invariant
geometries on $\mathrm{SU}(2)$ are uniformly doubling and give a detailed
estimate of the volume of balls that is valid for any of these geometries and
any radius. We discuss a number of consequences concerning the spectrum of the
associated Laplacians and the corresponding heat kernels.
