Non-variational extrema of exponential Teichm\"uller spaces

TL;DR

This paper investigates the existence of non-variational critical points in exponential Teichmüller spaces, revealing complex behaviors of mappings with p-exponentially integrable distortion and their invariance properties.

Contribution

It proves the existence of non-variational critical points in exponential Teichmüller spaces, highlighting new phenomena in the structure of these spaces.

Findings

Existence of non-variational critical points in $E_p$ spaces.

Mappings with p-exponentially integrable distortion are not invariant under all boundary-preserving diffeomorphisms.

Demonstrates complex behavior of extremal mappings in exponential Teichmüller 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.