LIMITS OF TEICHMULLER MAPS

作者：Tukia Pekka^*

来源：Journal d Analyse Mathematique, 2015, 125(1): 71-111.

DOI：10.1007/s11854-015-0003-7

Let Delta = {z is an element of C : vertical bar z vertical bar < 1}, <(Delta)over bar> = {z. C : vertical bar z vertical bar <= 1}, and Delta* = (C) over bar\(Delta) over bar = {z is an element of C : vertical bar z vertical bar > 1} boolean OR {infinity}. A Teichmuller map is a quasiconformal homeomorphism f of (C) over bar that is conformal outside of Delta* and such that, in Delta, the complex dilatation of f is of the form k (phi) over bar/vertical bar phi vertical bar, where 0 <= k 1 and phi is holomorphic. We consider sequences of such Teichmuller maps {f(j)} whose complex dilatations are of the form k(j)<(phi) over bar (j)/vertical bar phi(j)vertical bar, where phi(j) are holomorphic mappings, k(j) -> 1, and phi(j) tends to a holomorphic mapping. uniformly on compact subsets as j -> infinity. We assume that the L-1-norms of phi(j) and phi are uniformly bounded. If f(j) are suitably normalized, it is possible to pass to a subsequence such that fj tends to a conformal limit f outside (Delta) over bar. Since the f(j) are not uniformly quasiconformal, such a limit need not exist in (Delta) over bar. We show that there exists a subsequence of {f(j)} which tends to a modified form of a limit, called an extended limit, in (Delta) over bar. We construct a subsequence and an extended limit using a partition of (Delta) over bar, denoted D, whose elements are closed sets constructed from vertical trajectories of. as well as some closed arcs and points of partial derivative Delta. The extended limit, also denoted f, is defined on Delta* boolean OR D and satisfies a continuity condition called semicontinuity. The image f D = {f(X) : X is an element of D} is a family of closed sets o

• 出版日期2015-1