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 $\mathbb{C} \mathbb{P}^1$, 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 $\mathbb{C} \mathbb{P}^2$, is equal to zero.