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The Nitsche conjecture


Authors: Tadeusz Iwaniec, Leonid V. Kovalev and Jani Onninen
Journal: J. Amer. Math. Soc. 24 (2011), 345-373
MSC (2010): Primary 31A05; Secondary 58E20, 30C20
DOI: https://doi.org/10.1090/S0894-0347-2010-00685-6
Published electronically: November 10, 2010
MathSciNet review: 2748396
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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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Additional Information

Tadeusz Iwaniec
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244 and Department of Mathematics and Statistics, University of Helsinki, Finland
Email: tiwaniec@syr.edu

Leonid V. Kovalev
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244
MR Author ID: 641917
Email: lvkovale@syr.edu

Jani Onninen
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244
MR Author ID: 679509
Email: jkonnine@syr.edu

Keywords: Nitsche conjecture, harmonic mappings, minimal surfaces
Received by editor(s): November 3, 2009
Received by editor(s) in revised form: August 23, 2010
Published electronically: November 10, 2010
Additional Notes: The first author was supported by the NSF grant DMS-0800416 and the Academy of Finland grant 1128331.
The second author was supported by the NSF grant DMS-0913474.
The third author was supported by the NSF grant DMS-0701059.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.