The Exterior Green Function of the Mandelbrot Set: Conformal Construction and Quantitative Escape-Rate Approximation

For the quadratic family $f_c\left(z\right)=z^2+c$ , we study the exterior Green function of the Mandelbrot set $？$ through classical conformal uniformization, logarithmic potential theory, and quantitative escape-rate approximation. As classical input, we use the Douady–Hubbard parameter B？ttcher map ${\Phi }_{？}:？\backslash ？\to ？\backslash \bar{\mathbb{D}}$ , normalized at infinity, and the representation $G_{？}\left(c\right)=\log \left|{\Phi }_{？}\left(c\right)\right|$ . The harmonicity and positivity of $G_{？}$ on $？\backslash ？$ , its continuous extension by zero on $？$ , its normalized logarithmic growth at infinity, its uniqueness as the normalized exterior Green function, and the identity $G_{？}\left(c\right)=g_c\left(c\right)=2g_c\left(0\right)$ are classical facts recalled to fix the analytic framework and normalization. The quantitative escape-rate estimate used in this paper was established in the author’s earlier work. The contribution of the present paper is to provide a self-contained telescoping