Convex hull theorem for multiply connected domains in the plane with an estimate of the quasiconformal constant

For any multiply connected domain Ω in ℝ², let S be the boundary of the convex hull in H ³ of ℝ²Ω which faces Ω. Suppose in addition that there exists a lower bound l > 0 of the hyperbolic lengths of closed geodesics in Ω. Then there is always a K-quasiconformal mapping from S to Ω, which extends continuously to the identity on ∂S = ∂Ω, where K depends only on l. We also give a numerical estimate of K by using the parameter l. This work was supported by National Natural Science Foundation of China (Grant Nos. 10671004, 10831004) and the Doctoral Education Program Foundation of China (Grant No. 20060001003)