The Nitsche conjecture

Iwaniec, Tadeusz; Kovalev, Leonid; Onninen, Jani

doi:10.1090/S0894-0347-2010-00685-6

The Nitsche conjecture
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by Tadeusz Iwaniec, Leonid V. Kovalev and Jani Onninen

J. Amer. Math. Soc. 24 (2011), 345-373

DOI: https://doi.org/10.1090/S0894-0347-2010-00685-6

Published electronically: November 10, 2010

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Abstract:

The Nitsche conjecture is deeply rooted in the theory of doubly-connected minimal surfaces. However, it is commonly formulated in slightly greater generality as a question of existence of a harmonic homeomorphism between circular annuli \[ h \colon \mathbb A = A(r,R) \overset {\text {onto}}{\longrightarrow } A(r_\ast , R_\ast ) =\mathbb A^*. \] In the early 1960s, while attempting to describe all doubly-connected minimal graphs over a given annulus $\mathbb A^*$, J. C. C. Nitsche observed that their conformal modulus cannot be too large. Then he conjectured, in terms of isothermal coordinates, even more:

A harmonic homeomorphism $h\colon \mathbb {A} \overset {\text {onto}}{\longrightarrow } \mathbb {A}^\ast$ exists if and only if \[ \frac {R_\ast }{r_\ast } \geqslant \frac {1}{2} \left (\frac {R}{r}+ \frac {r}{R}\right ). \]

In the present paper we provide, among further generalizations, an affirmative answer to his conjecture.

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