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Alexander A. Borisenko

Publications and source records attributed to Alexander A. Borisenko.

14 recordsLinked to original sources
arXiv2025

Local rigidity of convex hypersurfaces in spaces of constant curvature

In this paper, we prove a local rigidity of convex hypersurfaces in the spaces of constant curvature of dimension $n\ge4$. Namely, we show that two convex isometric hypersurfaces are congruent locally around their corresponding under the isometry points of strict convexity. This result extends the result of E.P. Senkin, who showed such rigidity under the additional assumption of $C^1$-smoothness of the hypersurfaces.

math.DG↗
arXiv2024

Rigidity of convex hypersurfaces, Rigidity of convex hypersurfaces in multidimensional spaces of constant curvature

In 1972, E. P. Senkin generalized the celebrated theorem of A. V. Pogorelov on unique determination of compact convex surfaces by their intrinsic metrics in the Euclidean 3-space $E^3$ to higher dimensional Euclidean spaces $E^{n+1}$ under a mild assumption on the smoothness of the hypersurface. In this paper, we remove this assumption and thus establish this rigidity result for arbitrary compact, convex hypersurfaces in $E^{n+1}$, $n \ge 3$. We also prove the corresponding results in other model spaces of constant curvature.

math.DG↗
arXiv2023

Geodesic loops on tetrahedra in spaces of constant sectional curvature

Geodesic loops on polyhedra were studied only for Euclidean space and it was known that there are no simple geodesic loops on regular tetrahedra. Here we prove that: 1) On the spherical space, there are no simple geodesic loops on tetrahedra with internal angles $π/3 < α_i<π/2$ or regular tetrahedra with $α_i=π/2$, and there are three simple geodesic loops for each vertex of a tetrahedra with $α_i > π/2$ and the lengths of the edges $alpha_i>π/2$. 2) On the hyperbolic space, for every regular tetrahedron $T$ and every pair of coprime numbers $(p,q)$, there is one simple geodesic loop of $(p,q)$ type through every vertex of $T$.

math.DG↗
arXiv2021

Simple closed geodesics on regular tetrahedra in spherical space

On a regular tetrahedron in spherical space there exist the finite number of simple closed geodesics. For any pair of coprime integers $(p,q)$ it was found the numbers $α_1$ and $α_2$ depending on $p$, $q$ and satisfying the inequalities $π/3< α_1 < α_2 < 2π/3$ such that on a regular tetrahedron in spherical space with the faces angle $α\in \left( π/3, α_1 \right)$ there exists unique, up to the rigid motion of the tetrahedron, simple closed geodesic of type $(p,q)$, and on a regular tetrahedron with the faces angle $α\in \left( α_2, 2π/3 \right)$ there is no simple closed geodesic of type $(p,q)$.

math.MG↗
arXiv2019

Simple closed geodesics on regular tetrahedra in Lobachevsky space

We obtained a complete classification of simple closed geodesics on regular tetrahedra in Lobachevsky space. Also, we evaluated the number of simple closed geodesics of length not greater than $L$ and found the asymptotic of this number as $L$ goes to infinity.

math.MG↗
arXiv2009

Gaussian Mean curvature flow

We consider the evolution of a $n$-dimensional convex hypersurface in the euclidean space under mean curvature flow with densities $e^{\varepsilon \frac12 nμ^2 |x|^2}$, $\varepsilon =\pm 1$, and completely determine it depending on the relation between $μ$ and the upper or lower bound of the normal curvatures of the evolving hypersurface at time 0.

math.DG↗
arXiv2009

Mean curvature flow of graphs in warped products

Let $M$ be a complete Riemannian manifold which either is compact or has a pole, and let $φ$ be a positive smooth function on $M$. In the warped product $M\times_φ\mathbb R$, we study the flow by the mean curvature of a locally Lipschitz continuous graph on $M$ and prove that the flow exists for all time and that the evolving hypersurface is $C^\infty$ for $t>0$ and is a graph for all $t$. Moreover, under certain conditions, the flow has a well defined limit.

math.DG↗
arXiv2008

On the Global Structure of Hopf Hypersurfaces in Complex Space Form

It is known that a tube over a Kahler submanifold in a complex form is a Hopf hypersurface. In some sense the reverse statement is true: a connected compact generic immersed C^(2n-1) regular Hopf hypersurface in the complex projective plane is a tube iver an irreducible algebraic variety. In the complex hyperbolic space a connected compact generic immersed C^(2n-1) regular Hopf hypersurface is a geodesic hypersphere

math.DG↗
arXiv2007

Asymptotic Properties of Hilbert Geometry

We show that the spheres in Hilbert geometry have the same volume growth entropy as those in the Lobachevsky space. We give the asymptotic estimates for the ratio of the volume of metric ball to the area of the metric sphere in Hilbert geometry. Derived estimates agree with the well-known fact in the Lobachevsky space

math.DG↗