Leonid V. Kovalev

The Nitsche conjecture

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.

Hopf Differentials and Smoothing Sobolev Homeomorphisms

We prove that planar homeomorphisms can be approximated by diffeomorphisms in the Sobolev space

Diffeomorphic Approximation of Sobolev Homeomorphisms

W^{1,2}

Existence of Energy-Minimal Diffeomorphisms Between Doubly Connected Domains

and in the Royden algebra. As an application, we show that every discrete and open planar mapping with a holomorphic Hopf differential is harmonic.

The Reduced Module of the Complex Sphere

Every homeomorphism

Cofilin-1 and Other ADF/Cofilin Superfamily Members in Human Malignant Cells

{h : \mathbb X \to \mathbb Y}

Variation of quasiconformal mappings on lines

between planar open sets that belongs to the Sobolev class