A question of Kuhnau

TL;DR

This paper investigates whether long rectangles have associated holomorphic quadratic differentials and proves that they do not for any bounded convex quadrilateral, including rectangles, providing new insights into Teichmüller theory.

Contribution

It demonstrates the non-existence of holomorphic quadratic differentials for long rectangles and extends the result to all bounded convex quadrilaterals.

Findings

Long rectangles lack associated holomorphic quadratic differentials.

The non-existence result applies to all bounded convex quadrilaterals.

Provides a stronger result specifically for rectangles.

Abstract

It is well known that the square is not a Strebel's point (i.e., its extremal Beltrami coefficient is not Teichmuller). Many years ago, Reiner Kuhnau posed the question: does there exist in the case of a "long" rectangle the corresponding holomorphic quadratic differential? We prove that the answer is negative for any bounded convex quadrilateral and establish a stronger result for rectangles.