The Nitsche conjecture
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.
Keywords
Cite
@article{arxiv.0908.1253,
title = {The Nitsche conjecture},
author = {Tadeusz Iwaniec and Leonid V. Kovalev and Jani Onninen},
journal= {arXiv preprint arXiv:0908.1253},
year = {2011}
}
Comments
33 pages, 2 figures. Expanded introduction and references; added discussion of doubly-connected minimal surfaces