Department of Pure Mathematics

Founded in 2015

School of Mathematics and
Computer Sciences

V.N. Karazin Kharkiv
National University

Faculty

Vladimir . Dubovoy Full professor of the department of pure mathematics, doctor of sciences in mathematics

Sergey . Favorov Full professor of the department of pure mathematics, doctor of sciences in mathematics

Sergey . Gefter Associate professor of the department of pure mathematics, head of the department of pure mathematics, phd in mathematics

Vladimir (Volodymyr) M. Kadets Full professor of the department of pure mathematics, doctor of sciences in mathematics

Dmitry . Shepelsky Doctor of sciences in mathematics

Alexander L. Yampolsky Professor of the department of pure mathematics, doctor of sciences in mathematics

Vasyl O. Gorkavyy Doctor of sciences in mathematics, associate professor

Alexander V. Rezounenko Professor of the department of pure mathematics, doctor of sciences in mathematics

Anna M. Vishnyakova Professor of the department of pure mathematics, doctor of sciences in mathematics

Tamara . Fastovska Associate professor of the department of pure mathematics, phd in mathematics, associate professor

Nataliуa . Girya Associate professor of the department of pure mathematics, phd in mathematics

Eugene . Karolinsky Associate professor of the department of pure mathematics, phd in mathematics

Eugene V. Petrov Phd in mathematics, senior lecturer

Dmytro . Seliutin Ph.d.

Aleksey . Shcherbina Phd in mathematics, senior lecturer

Olesia O. Zavarzina Ph.d., associate professor of the department of pure mathematics, associate professor

Viktoria V. Davydova Leading engineer

Anna . Goncharuk

Iryna V. Kats Leading engineer

petrov@karazin.ua

Eugene V. Petrov

Phd in mathematics, senior lecturer

Link on external publications: ResearchGate.

List of selected publications

E.V. Petrov The Gauss Map of Submanifolds in the Heisenberg Group // Differential Geometry and its Applications, 2011, vol. 29, p. 516–532, 2011

We obtain criteria for the harmonicity of the Gauss map of submanifolds in the Heisenberg group and consider the examples demonstrating the connection between the harmonicity of this map and the properties of the mean curvature field. Also, we introduce a natural class of cylindrical submanifolds and prove that a constant mean curvature hypersurface with harmonic Gauss map is cylindrical.

Keywords: Heisenberg group, Gauss map, harmonic map, mean curvature field, constant mean curvature hypersurface

Ye.V. Petrov Submanifolds with the Harmonic Gauss Map in Lie Groups // Journ. of Math. Phys., An., Geom., 4 (2008), no. 2, p. 278-293, 2008

In this paper we find a criterion for the Gauss map of an immersed smooth submanifold in some Lie group with left invariant metric to be harmonic. Using the obtained expression we prove some necessary and sufficient conditions for the harmonicity of this map in the case of totally geodesic submanifolds in Lie groups admitting biinvariant metrics. We show that, depending on the structure of the tangent space of a submanifold, the Gauss map can be harmonic in all biinvariant metrics or non-harmonic in some metric. For $2$-step nilpotent groups we prove that the Gauss map of a geodesic is harmonic if and only if is constant.

Keywords: Left invariant metric, biinvariant metric, Gauss map, harmonic map, $2$-step nilpotent group, totally geodesic submanifold

Ye.V. Petrov The Gauss Map of Hypersurfaces in 2-Step Nilpotent Lie Groups // Journ. of Math. Phys., An., Geom., 2 (2006), no. 2, p. 186-206, 2006

In this paper we consider smooth oriented hypersurfaces in $2$-step nilpotent Lie groups with a left invariant metric and derive an expression for the Laplacian of the Gauss map for such hypersurfaces in the general case and in some particular cases. In the case of CMC-hypersurface in the $2m+1$-dimensional Heisenberg group we also derive necessary and sufficient conditions for the Gauss map to be harmonic and prove that for $m=1$ all CMC-surfaces with the harmonic Gauss map are ''cylinders''.

Keywords: 2-step nilpotent Lie group, Heisenberg group, left invariant metric, Gauss map, harmonic map, minimal submanifold, constant mean curvature