The Nitsche conjecture

TL;DR

This paper proves Nitsche's conjecture, establishing a condition for the existence of harmonic homeomorphisms between annuli, and identifies the catenoid's upper slab as having maximal conformal modulus among minimal graphs over an annulus.

Contribution

The paper provides a proof of Nitsche's conjecture and characterizes the maximal conformal modulus for minimal graphs over annuli.

Findings

Confirmed Nitsche's conjecture with an affirmative proof.

Identified the upper slab of the catenoid as having the greatest conformal modulus.

Established a necessary inequality for harmonic homeomorphisms between annuli.

Abstract

The conjecture in question concerns the existence of a harmonic homeomorphism between circular annuli A(r,R) and A(r*,R*), and is motivated in part by the existence problem for doubly-connected minimal surfaces with prescribed boundary. In 1962 J.C.C. Nitsche observed that the image annulus cannot be too thin, but it can be arbitrarily thick (even a punctured disk). Then he conjectured that for such a mapping to exist we must have the following inequality, now known as the Nitsche bound: R*/r* is greater than or equal to (R/r+r/R)/2. In this paper we give an affirmative answer to his conjecture. As a corollary, we find that among all minimal graphs over given annulus the upper slab of catenoid has the greatest conformal modulus.