On canonically polarized Gorenstein 3-folds satisfying the Noether equality

We study canonically polarized Gorenstein $3$-folds with at most terminal singularities and satisfying $K_X^3=\frac 43p_g(X)-\frac {10}3$ and $p_g(X) \ge 7$. We characterize the canonical maps of such $3$-folds, describe a structure theorem for the locally factorial ones and completely classify the smooth ones. New examples of canonically polarized smooth $3$-folds with $K_X^3=\frac 43p_g(X)-\frac {10}3$ and $p_g(X) \ge 7$ are constructed. These examples are natural extensions of those constructed by M.~Kobayashi.