Analytic Capacity and Holomorphic Motions

The paper proves that for a holomorphic motion f with each slice f_λ holomorphic on Ĉ\K and fixing ∞, the function λ ↦ log γ(K_λ) is harmonic, by showing γ(K)/γ(K_λ) equals a holomorphic, nonvanishing integral built from f(λ,·), hence its log-modulus is harmonic (Theorem 1.1 and its proof) . The candidate’s solution proves harmonicity via the same core mechanism: identify the coefficient a(λ) at ∞ for f_λ, use conformal naturality of analytic capacity under mappings of complements to get γ(K_λ) proportional to γ(K), and conclude that log γ(K_λ) = const ± Re log a(λ) is harmonic. Aside from a minor misattribution (they ascribe conformality on Ĉ\K to the λ-lemma rather than to the paper’s explicit assumption) and a normalization/sign convention on the capacity-scaling factor versus a(λ) that does not affect harmonicity, the arguments materially coincide with the paper’s proof.