A Geometric Interpretation of Half-Plane Capacity

TL;DR

This paper provides a geometric interpretation of half-plane capacity by relating it to a measure based on tangent balls, offering bounds that connect probabilistic and geometric perspectives in complex analysis.

Contribution

It introduces a new geometric quantity, hsiz(A), and establishes bounds linking it to the half-plane capacity, enhancing understanding of SLE-related conformal invariants.

Findings

hcap(A) is comparable to hsiz(A) within explicit bounds

Established inequalities: hsiz(A)/66 < hcap(A) ≤ 7 hsiz(A)/(2π)

Connects probabilistic capacity with geometric measure in the upper half-plane

Abstract

Let A be a bounded, relatively closed subset of the upper half plane H whose complement C is simply connected. If B_t is a standard complex Brownian motion starting at iy and t_A = inf {t > 0: B_t not in C}, the half-plane capacity of A, hcap(A) is defined to be the limit as y goes to infinity of y E[Im(B_{t_A}]. This quantity arises naturally in the study of Schramm-Loewner Evolutions (SLE). In this note, we show that hcap(A) is comparable to a more geometric quantity hsiz(A) that we define to be the 2-dimensional Lebesgue measure of the union of all balls tangent to R whose centers belong to A. Our main result is that hsiz(A)/66 < hcap(A) leq 7 hsiz(A)/(2 pi).