In this chapter, we use the fruits of the, already proven, existence of Sullivan conformal measures with a minimal exponent and its various dynamical characterizations. Having compact nonrecurrence, we are able to prove in this chapter that this minimal exponent is equal to the Hausdorff dimension $\HD(J(f))$ of the Julia set $J(f)$, which we denote by $h$. We also obtain here some strong restrictions on the possible locations of atoms of such conformal measures. In the last section of this chapter, we culminate our work on Sullivan conformal measures for elliptic functions treated on their own. There and from then onward, we assume that our compactly nonrecurrent elliptic function is regular (we define this concept). For this class of elliptic functions, we prove the uniqueness and atomlessness of $h$-conformal measures along with their first fundamental stochastic properties such as ergodicity and conservativity.