  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.
