The Nitsche conjecture

Details The Nitsche conjecture The Nitsche conjecture

2009-08-10

arxiv.org/abs/0908.1253v3

J. Amer. Math. Soc. 24 (2011), no. 2, 345-373

Mathematics

Complex Variables

33 pages, 2 figures. Expanded introduction and references; added discussion of doubly-connected minimal surfaces

Scientific paper

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.

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