Mappings of finite distortion: formation of cusps

Mappings of finite distortion: Formation of cusps III

“…For the basic properties of these mappings see [1,2,5,6,8]. The purpose of this paper is to continue our efforts [7,9] to understand the geometry of the image of the unit disk under a homeomorphism with a nicely integrable distortion function.…”

“…Based on our previous work [7,9] we know essentially sharp (exponential integrability) conditions on the distortion function under which Ω s can arise as the image of the unit disk B := B(0, 1) under a homeomorphism of finite distortion.…”

Mappings of finite distortion: Formation of cusps II

“…Thus if we have a conformal or quasiconformal mapping f : B → Ω s , it does not extend to a quasiconformal mapping of the entire plane. In [8] it was shown that if the extension is allowed to be a homeomorphism of finite distortion, then such extension may exist, but in such a case λ ≤ 1/s if exp(λK) ∈ L 1 loc (R 2 ). In this paper we will consider the situation when f : B → Ω s is only a homeomorphism of finite distortion to begin with.…”

“…This construction follows the method presented in [8], Theorem 1; we only need to fine-tune the definitions of g(r) and G s (r) there to give us the correct result. For the convenience of the reader we represent the entire construction.…”

“…Regarding these domains, we have the following theorem: If f is additionally quasiconformal in the unit disc B, the critical λ is 1/s (see [8]). Based on the constructions we have attempted to make, we expect that the critical λ in Theorem 1 should be 2/s, but we have not been able to prove this.…”