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Alexander Tumanov

Publications and source records attributed to Alexander Tumanov.

At least 19 recordsLinked to original sources
arXiv2025

Realizing semisimple Lie groups as holomorphic automorphism groups of bounded domains

We consider a problem whether a given Lie group can be realized as the group of all biholomorphic automorphisms of a bounded domain in the affine complex space. In an earlier paper of 1990, we proved the result for connected linear Lie groups. In this paper we prove the result for a fairly large class of Lie groups including all connected real semisimple groups. This class contains many non-linear Lie groups.

math.CV↗
arXiv2023

H\"older regularity of the $\bar\partial-$equation on the polydisc

In this note, we show the existence of a solution operator to the $\bar\partial-$equation in the polydisc that preserves H\"older regularity. This solution operator is constructed using Henkin's formula. It is a well-known fact that solution operators to the $\bar\partial-$equation on product domains do not improve H\"older regularity. Hence, this solution operator is optimal in that regard.

math.CV↗
arXiv2022

Infinitesimal automorphisms of quadrics and second jet determination for CR mappings

We consider a problem whether a CR mapping of a generic manifold in complex space is uniquely determined by its finite jet at a point, which is referred to as finite jet determination. We derive the finite jet determination for CR mappings of smooth Levi nondegenerate manifolds of arbitrary codimension from the finite dimensionality of the algebras of infinitesimal automorphisms of the corresponding quadrics. Previously, this implication was known for real analytic manifolds. We prove a new 2-jet determination result that covers most affirmative results on this matter obtained so far.

math.CV↗
arXiv2020

Stationary discs and finite jet determination for CR mappings in higher codimension

We discuss stationary discs for generic CR manifolds and apply them to the problem of finite jet determination for CR mappings. We prove that a CR diffeomorphism of two finitely smooth strictly pseudoconvex Levi generating CR manifolds is uniquely determined by its 2-jet at a given point. A new key element of the proof is the existence of non-defective stationary discs.

math.CV↗
arXiv2017

Scarcity of periodic orbits in outer billiards

We give a simple proof of our previous result with V. Zharnitsky that the set of period 4 orbits in planar outer billiard with piecewise smooth convex boundary has empty interior, provided that no four corners of the boundary form a parallelogram. We also obtain results on period 5 and 6 orbits.

math.DS↗
arXiv2016

Symplectic non-squeezing in Hilbert space and discrete Schrödinger equations

We prove a generalization of Gromov's symplectic non-squeezing theorem for the case of Hilbert spaces. Our approach is based on filling almost complex Hilbert spaces by complex discs partially extending Gromov's results on existence of $J$-complex curves. We apply our result to the flow of the discrete nonlinear Schrödinger equation.

math.SG↗
arXiv2015

Pseudoholomorphic discs and symplectic structures in Hilbert space

We develop the theory of $J$-holomorphic discs in Hilbert spaces with almost complex structures. As an aplication, we prove a version of Gromov's symplectic non-squeezing theorem for Hilbert spaces. It can be applied to short-time symplectic flows of a wide class of Hamiltonian PDEs.

math.CV↗
arXiv2012

Minimal biquadratic energy of 5 particles on 2-sphere

Consider n points on the unit 2-sphere. The potential of the interaction of two points is a function f(r) of the distance r between the points. The total energy E of n points is the sum of the pairwise energies. The question is how to place the points on the sphere to minimize the energy E. For the Coulomb potential f(r)=1/r, the problem goes back to Thomson (1904). The results for n < 5 are well known. We focus on the case n=5, which turns out to be difficult. In this case, the following results have been obtained. For n=5, Dragnev, Legg, and Townsend (2002) give a solution of the problem for f(r)=-log r known as Whyte's problem. Hou and Shao (2009) give a rigorous computer-aided solution for f(r)=-r. Schwartz (2010) gives a rigorous computer-aided solution of Thompson's problem. We give a solution for biquadratic potentials.

math.MG↗
arXiv2008

Filling real hypersurfaces by pseudoholomorphic discs

We study pseudoholomorphic discs with boundaries attached to a real hypersurface in an almost complex manifold. We give sufficient conditions for filling a one sided neighborhood of the hypersurface by the discs.

math.CV↗
arXiv2008

Analytic continuation from a family of lines

Given a function f in the exterior of a convex curve in the real plane, we prove that if the restrictions of f to the tangent lines to the curve extend as entire functions, then the function f is an entire function of two variables.

math.CV↗
arXiv2006

Periodic orbits in outer billiards

It is shown that the set of 4-period orbits in outer billiard with piecewise smooth convex boundary has an empty interior, provided that no four corners of the boundary form a parallelogram.

math.DS↗