Geometry Research Group

V.N. Karazin Kharkiv
National University

Склад секції

Alexander L. Yampolsky Docent (associate professor) of department of fundamental mathematics, z, doctor of sciences in physics and mathematics, docent (associate professor)

Kostiantyn D. Drach Phd in mathematics

Olga V. Lykova Phd in mathematics, senior lecturer

Olena M. Nevmerzhitska Phd in mathematics, senior lecturer

Eugene V. Petrov Phd in mathematics, senior lecturer

Olena O. Shugailo Phd in mathematics, senior lecturer

Dmytro V. Bolotov Doctor of sciences in physics and mathematics, senior researcher of iltpe

Vasyl O. Gorkavyy Doctor of sciences in physics and mathematics, docent (associate professor)

. Docent (associate professor) of department of theoretical and applied computer science , phd in mathematics, docent (associate professor)

P. G. Dolya Docent (associate professor) of department of theoretical and applied computer science , phd for industries

Iryna V. Kats Lead engineer

Dm. I. Vlasenko Phd in mathematics, senior lecturer

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A.L. Yampolsky

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Eugene V. Petrov

Phd in mathematics, senior lecturer

Link on external publications: ResearchGate.

List of selected publications

A.A. Borisenko Surfaces in the Three-Dimensional Heisenberg Group on Which the Gauss Map Has Bounded Jacobian // Mathematical Notes, 89 (2011), p. 746-748, 2011

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

E.V. Petrov On the Gauss Map of Submanifolds in Lie Groups (in Russian) // Dopovidi NANU, 2008, 11, p. 28-31, 2008

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

L.A. Masal'tsev, E.V. Petrov On the Stability of Minimal Surfaces in the 3-Dimensional Heisenberg Group (in Russian) // Visnyk KhNU, 645, 2004, p. 135-141, 2004

L.A. Masal'tsev, E.V. Petrov Minimal Submanifolds in the Heisenberg Group (in Russian) // Visnyk KhNU, 602, 2003, p. 35-45, 2003