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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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The Nitsche conjecture
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by Tadeusz Iwaniec, Leonid V. Kovalev and Jani Onninen PDF
J. Amer. Math. Soc. 24 (2011), 345-373 Request permission

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.

References
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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
  • 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.
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2748396