Abstract Letϕ: C→Cbe a bilipschitz map. We prove that ifE⊂Cis compact, andγ(E),α(E) stand for its analytic and continuous analytic capacity respectively, thenC ư1γ(E)≤γ(ϕ(E))≤Cγ(E) andCư1α(E)≤α(ϕ(E))≤Cα(E), whereCdepends only on the bilipschitz constant ofϕ. Further, we show that if µis a Radon measure on Cand the Cauchy transform is bounded onL2(µ), then the Cauchy transform is also bounded onL2(ϕµ), whereϕµis the image measure ofµbyϕ. To obtain these results, we estimate the curvature of ϕµ by means of a corona type decomposition. 1. Introduction 2. Preliminaries 3. The corona decomposition 4. Construction of the curvesΓR, R∈Top(E) 5. The packing condition for the top squares 6. Estimates for the high curvature squares 7. Estimates for the low density squares 8. The curvature ofϕµ