A weighted extremal function and equilibrium measure

TL;DR

This paper derives an explicit formula for a weighted extremal function associated with a convex body in complex space, computes its Monge-Ampère measure, and determines the Alexander capacity of real projective space.

Contribution

It provides a new explicit formula for the weighted extremal function $V_{K,Q}$ using extremal functions and algebraic characterization, advancing potential theory in complex analysis.

Findings

Explicit formula for $V_{K,Q}$ in terms of $z$ and $|z|$

Calculation of Alexander capacity $T_{ ext{Alexander}}(f R P^n) = 1/ oot 2 ext{ of } 2$

Determination of the Monge-Ampère measure of $V_{K,Q}$

Abstract

Let $K={\bf R}^n\subset {\bf C}^n$ and $Q(x):=\frac{1}{2}\log (1+x^2)$ where $x=(x_1,...,x_n)$ and $x^2 = x_1^2+\cdots +x_n^2$. Utilizing extremal functions for convex bodies in ${\bf R}^n\subset {\bf C}^n$ and Sadullaev's characterization of algebraicity for complex analytic subvarieties of ${\bf C}^n$ we prove the following explicit formula for the weighted extremal function $V_{K,Q}$: $$V_{K,Q}(z)=\frac{1}{2}\log \bigl( [1+|z|^2] + \{ [1+|z|^2]^2-|1+z^2|^2\}^{1/2})$$ where $z=(z_1,...,z_n)$ and $z^2 = z_1^2+\cdots +z_n^2$. As a corollary, we find that the Alexander capacity $T_{\omega}({\bf R} {\bf P}^n)$ of ${\bf R} {\bf P}^n$ is $1/\sqrt 2$. We also compute the Monge-Amp\`ere measure of $V_{K,Q}$: $$(dd^cV_{K,Q})^n = n!\frac{1}{(1+x^2)^{\frac{n+1}{2}}}dx.$$