Non-variational extrema of exponential Teichm\"uller spaces

Abstract

The exponential Teichm\"uller spaces $E_p$, $0\leq p \leq \infty$, interpolate between the classical Teichm\"uller space ($p=\infty$) and the space of harmonic diffeomorphisms $(p=0)$. In this article we prove the existence of non-variational critical points for the associated functional: mappings $f$ of the disk whose distortion is $p$-exponentially integrable, $0<p<\infty$, yet for {\em any} diffeomorphism $g(z)$ of $\ID$ with $g|\partial\ID=identity$ and $g\neq identity$ we have $f\circ g$ is not of $p$-exponentially integrable distortion.