Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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
Published electronically: November 10, 2010
MathSciNet review: 2748396
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

$\displaystyle 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

$\displaystyle \frac{R_\ast}{r_\ast} \ge \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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 31A05, 58E20, 30C20

Retrieve articles in all journals with MSC (2010): 31A05, 58E20, 30C20


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
Email: lvkovale@syr.edu

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

DOI: http://dx.doi.org/10.1090/S0894-0347-2010-00685-6
PII: S 0894-0347(2010)00685-6
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.