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

TL;DR

This paper investigates properties of (weak) m-extremals and m-geodesics across various complex domains, establishing new results on their equivalence, boundary behavior, and providing examples of extremals that are not geodesics.

Contribution

It demonstrates the equivalence of weak and strong m-extremality in certain convex complex ellipsoids and provides the first examples of 3-extremals that are not 3-geodesics in convex domains.

Findings

Proves 3-geodesity of 3-extremals in the Euclidean ball.

Shows equivalence of weak and strong m-extremality in specific convex ellipsoids.

Provides examples of 3-extremals that are not 3-geodesics in convex domains.

Abstract

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.