On the energy of quasiconformal mappings and pseudoholomorphic curves in complex projective spaces

Abstract

We prove that the energy density of uniformly continuous, quasiconformal mappings, omitting two points in CP, is equal to zero. We also prove the sharpness of this result, constructing a family of uniformly continuous, quasiconformal mappings, whose areas grow asymptotically quadratically. Finally, we prove that the energy density of pseudoholomorphic Brody curves, omitting three “complex lines” in general position in CP, is equal to zero. According to the Picard Theorem, a holomorphic function f defined on the complex plane C is constant as soon as f(C) omits at least three values in CP1. This result has different generalizations in at least two directions. S.Rickman [11] proved that for every n ≥ 2 and for every K > 1, a nonconstant entire Kquasiregular mapping in Rn omits at most m = m(n,K) values. M.Green [6] proved that a holomorphic map from C to the complex projective space CPn, omitting (2n+ 1) hyperplanes in general position, is constant. An almost complex version of that result was proved by J.Duval [5] for entire pseudoholomorphic curves in the complement of five J-lines, in general position in CP2 endowed with an almost complex structure J tamed by the Fubini Study metric ωFS . Let f be a mapping defined on C with values in CPn, f ∈ W 1,2 loc (C). We recall that if D ⊂⊂ C, then Area(f(D)) := ∫ D f ωFS is the area of f(D), counted with multiplicity. Then, the energy density E(f) defined by E(f) = lim sup R→∞ 1 πR2 Area(f(DR)) = lim sup R→∞ 1 πR2 ∫