Harmonic mappings of an annulus, Nitsche conjecture and its generalizations

TL;DR

This paper proves Nitsche's conjecture on harmonic homeomorphisms between annuli for certain annulus widths, introduces differential operators for forced evolution, and establishes bounds on the ratio of outer to inner radii.

Contribution

It confirms Nitsche's conjecture for annuli with log ratio up to 3/2 and extends results to forced harmonic evolutions using new differential operator techniques.

Findings

Proves Nitsche's conjecture for $ ext{log}(R/r) \\leq 3/2$.

Establishes bounds for forced harmonic evolution.

Introduces differential operators for circular means in harmonic mappings.

Abstract

As long ago as 1962 Nitsche conjectured that a harmonic homeomorphism $h \colon A(r,R) \to A(r_*, R_*)$ between planar annuli exists if and only if $\frac{R_*}{r_*} \ge {1/2} (\frac{R}{r} + \frac{r}{R})$. We prove this conjecture when the domain annulus is not too wide; explicitly, when $\log \frac{R}{r} \le {3/2}$. For general $A(r,R)$ the conjecture is proved under additional assumption that either $h$ or its normal derivative have vanishing average on the inner boundary circle. This is the case for the critical Nitsche mapping which yields equality in the above inequality. The Nitsche mapping represents so-called free evolution of circles of the annulus $A(r,R)$. It will be shown on the other hand that forced harmonic evolution results in greater ratio $\frac{R_*}{r_*}$. To this end, we introduce the underlying differential operators for the circular means of the forced evolution and use them to obtain sharp lower bounds of $\frac{R_*}{r_*}$.