(Weak) $m$-extremals and $m$-geodesics

We present a collection of results on (weak) $m$-extremals and $m$-geodesics, concerning general properties, the planar case, quasi-balanced pseudoconvex domains, complex ellipsoids, the Euclidean ball and boundary properties. We prove $3$-geodesity of $3$-extremals in the Euclidean ball. Equivalence of weak $m$-extremality and $m$-extremality in some class of convex complex ellipsoids, containing symmetric ones and ${\mathcal{C}}^2$-smooth ones is showed. Moreover, first examples of $3$-extremals being not $3$-geodesics in convex domains are given.