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 ℂℙ¹, 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 ℂℙ², is equal to zero.