The Nitsche conjecture
By Tadeusz Iwaniec, L. Kovalev, Jani Onninen. 2009
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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 : A = A(r, R) onto −→ A(r∗, R∗) = A In the early 1960s, while attempting to describe all doubly connected minimal graphs over a given annulus 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 : A onto −→ A∗ exists if an only if: R∗ r∗ > 1 2 „ R r + r R « This fascinating and engaging problem remained open for almost a half of a century. In the present paper we provide, among further generalizations, an affirmative answer to his conjecture.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/25ccd81c3386df7175717c4bb53329278c07f03f
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Existence of energy-minimal diffeomorphisms between doubly connected domains (2010, background): “A∗ is the image of Ωunder the extremal Nitsche mapping h∗(z) = 1 2 z+ 1 ¯z which is not only harmonic but also energy-minimal in H1,2(Ω,A∗) [2, Corollary 2]. The uniqueness part of Theorem 1.1 in [14] states that h∗ is the unique harmonic homeomorphism from Ω onto A∗, up to a conformal automorphis”