Conformal Laminations on the Circle

A lamination L on T is an equivalence relation on T. In this paper, we consider laminations L(φ) induced by some continuous mapping φ: D → C which is one-to-one in D, i.e ξ L(φ) ƞ if and only if φ(ξ) = φ(ƞ) for any ξ, ƞ ∈ T . The laminations arise as in above can be characterized topologically as continuous and flat laminations. Our major question is what conditions on L can ensure that L=L(φ) for some continuous mapping φ: D → C which is conformal in D. In this paper, we study various aspects of conformal laminations and get conditions for conformality in several configurations. We relate the conformal welding problem with the classical conformal se,ving problem. By the extremal length method, we obtain a generalization of Oikawa's condition for conformal sewings to a sufficient condition for conformal laminations and also obtain necessary conditions for conformal laminations. We prove that a continuous, flat lamination L on T is conformal if capML=0 and the quotient space ML/L is a totally disconnected set where ML is the set of multiple points of L. Let E be a compact subset of T. Suppose that In = (an, bn) are the components of the set T\E . We define L to be the lamination that identifies an and bn for each n. We prove that if capE > 0 and E is Dirichlet regular, then the lamination L as described above is conformal. Furthermore, the quotient space E/L is homeomorphic to the unit circle. We also conjecture that: If cap ML = 0, then L is conformal.