Alexander Vasil'ev

On the geometry of Hele-Shaw flows with small surface tension

We discuss the Hele‐Shaw problem in the plane in two basic cases. The first one deals with the classical situation of injection through a unique source in a finite region. The second one is concerned with the free boundary extending to the point at infinity. Starting with the earlier works of L. A. Galin [9] and P. Ya. Polubarinova-Kochina [19, 20], various aspects of the planar Hele‐Shaw viscous flows with vanishing surface tension were investigated by a number of scientists. It is known [31] that in the zero-surface-tension Hele‐Shaw problem with an initial region with an analytic boundary the classical solution exists locally in time. Recently [25], it became clear that the model could be interpreted as a particular case of the abstract Cauchy problem, and thus, the classical solvability (locally in time) may be proved using the nonlinear abstract Cauchy‐Kovalevskaya Theorem.

Noded Teichmüller spaces

LetG be a finitely generated Kleinian group and let Δ be an invariant collection of components in its region of discontinuity. The Teichmüller spaceT(Δ,G) supported in Δ is the space of equivalence classes of quasiconformal homeomorphisms with complex dilatation invariant underG and supported in Δ. In this paper we propose a partial closure ofT(Δ,G) by considering certain deformations of the above hemeomorphisms. Such a partial closure is denoted byNT(Δ,G) and called thenoded Teichmüller space ofG supported in Δ. Some concrete examples are discussed.

Digons and angular derivatives of analytic self-maps of the unit disk

We present a geometric approach to a well-known sharp inequality, due to Cowen and Pommerenke, about angular derivatives of general univalent self-maps of the unit disk.

FINITE DIMENSIONAL GRADING OF THE VIRASORO ALGEBRA

Abstract The Virasoro algebra is a central extension of the Witt algebra, the complexified Lie algebra of the sense preserving diffeomorphism group of the circle Diff 𝑆1. It appears in Quantum Field Theories as an infinite dimensional algebra generated by the coefficients of the Laurent expansion of the analytic component of the momentum-energy tensor, Virasoro generators. The background for the construction of the theory of unitary representations of Diff 𝑆1 is found in the study of Kirillovs manifold Diff 𝑆1=𝑆1. It possesses a natural Kählerian embedding into the universal Teichmüller space with the projection into the moduli space realized as an infinite-dimensional body of the coefficients of univalent quasiconformally extendable functions. The differential of this embedding leads to an analytic representation of the Virasoro algebra based on Kirillovs operators. In this paper we overview several interesting connections between the Virasoro algebra, Teichmüller theory, Löwner representation of univalent functions, and propose a finite-dimensional grading of the Virasoro algebra such that the grades form a hierarchy of finite dimensional algebras which, in their turn, are the first integrals of Liouville partially integrable systems for coefficients of univalent functions.