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

Tomasz Warszawski

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.

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

Abstract

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

Paper Structure

This paper contains 10 sections, 33 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Idea of (weak) $\boldsymbol m$-extremals and $\boldsymbol m$-geodesics
  3. Notation and definitions
  4. Main results
  5. General properties and the planar case
  6. Quasi-balanced pseudoconvex domains
  7. Complex ellipsoids
  8. The Euclidean ball
  9. Boundary properties
  10. List of problems

Key Result

Lemma 2.1

Let $D\subset\mathbb{C}^n$ be a domain and let $\lambda_1,\ldots,\lambda_m\in\mathbb{D}$ be different points.

Theorems & Definitions (75)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Lemma 3.1
  • proof
  • ...and 65 more