Sobolev Homeomorphisms and Brennan’s Conjecture

Abstract

Let \(\Omega \subset {\mathbb {R}}^n\) be a domain that supports the \(p\)-Poincare inequality. Given a homeomorphism \(\varphi \in L^1_p(\Omega )\), for \(p>n\) we show that the domain \(\varphi (\Omega )\) has finite geodesic diameter. This result has a direct application to Brennan’s conjecture and quasiconformal homeomorphisms. The Inverse Brennan’s conjecture states that for any simply connected plane domain \(\Omega ' \subset {\mathbb {C}}\) with non-empty boundary and for any conformal homeomorphism \(\varphi \) from the unit disc \({\mathbb {D}}\) onto \(\Omega '\) the complex derivative \(\varphi '\) is integrable in the degree \(s, -2 2\) is not possible for domains \(\Omega '\) with infinite geodesic diameter.