Aleksander Andreevich Borisenko

Aleksander Andreevich Borisenko
Sumy State University | SSU · Department of Mathematical Analysis and Optimization

Professor

About

136
Publications
8,055
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596
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November 1972 - September 2012
V. N. Karazin Kharkiv National University
Position
  • Heard of Geometry Department

Publications

Publications (136)
Article
We prove a reverse isoperimetric inequality for domains homeomorphic to a disc with the boundary of curvature bounded below lying in two-dimensional Alexandrov spaces of curvature c. We also study the equality case.
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We provide the bounds on the total curvature of curves in non-symmetric Minkowski spaces with respect to the Euclidean counterpart. This result allows us to generalize Fenchel and Fary-Milnor theorems about curves in the Euclidean space. We also provide an upper bound for lengths of closed curves that are contained in a Minkowski ball of a fixed ra...
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We consider Finsler submanifolds in Minkowski spaces, and in particular, in Randers spaces. We give generalizations of the Toponogov and Cheeger-Gromoll theorems to the case of Randers spaces. Sufficient conditions are obtained for complete Finsler submanifolds in Minkowski spaces to be cylindrical. We also find conditions under which the convexity...
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We give a sharp lower bound on the area of the domain enclosed by an embedded curve lying on a two-dimensional sphere provided that geodesic curvature of this curve is bounded from below. Furthermore, we prove some dual inequalities for convex curves whose curvatures are bounded from above.
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For embedded closed curves with curvature bounded below, we prove an isoperimetric inequality estimating the minimal area bounded by such curves for a fixed perimeter.
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For a convex domain $D$ that is enclosed by the hypersurface $\partial D$ of bounded normal curvature, we prove an angle comparison theorem for angles between $\partial D$ and geodesic rays starting from some fixed point in $D$, and the corresponding angles for hypersurfaces of constant normal curvature. Also, we obtain a comparison theorem for sup...
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In this chapter we investigate the convergence of the mean curvature flow of submanifolds in Euclidean and hyperbolic spaces with Gaussian density. For Euclidean case, we prove that the flow deforms a closed submanifold with pinching condition to a “round point” in finite time.
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For a Riemannian manifold and a compact domain bounded by a hypersurface with normal curvature bounded below, estimates are obtained in terms of the distance from to for the angle between the geodesic line joining a fixed interior point in to a point on and the outward normal to the surface. Estimates for the width of a spherical shell containing s...
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For a Riemannian manifold M^{n+1} and a compact domain \Omega in M^{n+1} whose boundary $\partial \Omega$ is a hypersurface with the normal curvature bounded from below we give a sharp estimate for the angle between a geodesic ray from a fixed point O inside the domain \Omega through a point on $\partial \Omega$ and the outward normal to $\partial...
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We consider simple closed curves in a Minkowski space. We give bounds of the total Minkowski curvature of the curve in terms of the total Euclidean curvature and of normal curvatures on the indicatrix (supposed to be a central symmetric hypersurface) of the Minkowski norm. Corollaries of this result provide analogues to Fenchel and Fary-Milnor theo...
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We prove that the normal curvatures of hyperspheres, the Rund curvature, and the Finsler curvature of circles in Hilbert geometry tend to 1 as the radii tend to infinity
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T -кривизной. Доказывается, что при определенных ограничениях на нормальную кривизну такие гиперповерхности являются выпуклыми, вложенными, гомеоморфными сфере. Для этого доказывается обобщение теоремы Рауха для экспоненциального отображения относительно гиперповерхности и показывается выпуклость параллельных гиперповерхностей.
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[2].However, this result can be strengthened; moreover, it can be extended to nilpotent groups of nilpotencyclass 2, and in the cases where such a group admits a locally isometric immersion in Euclidean space,the codimension of the immersion can be estimated. Even more, Riemannian manifolds curvature-equivalent to nilpotent groups of class 2with le...
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It is given the survey of comparison theorems for volumes balls and spheres in Finsler and Hilbert spaces.
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In the Euclidean Space R(n+1) with a density e(epsilon 1/2n mu 2|x|2), (epsilon = +/- 1), we consider the flow of a hypersurface driven by its mean curvature associated to this density. We give a detailed account of the evolution of a convex hypersurface under this flow. In particular, when epsilon = -1 (Gaussian density), the hypersurface can expa...
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Let $M$ be a complete Riemannian manifold which either is compact or has a pole, and let $\varphi$ be a positive smooth function on $M$. In the warped product $M\times_\varphi\mathbb R$, 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 hypersurfa...
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Life and the mathematical legacy of the great mathematician A.V. Pogorelov.
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Life and the mathematical legacy of the great mathematician A.V. Pogorelov.
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We give upper and lower bounds for the ratio of the volume of metric ball to the area of the metric sphere in Finsler-Hadamard manifolds with pinched S-curvature. We apply these estimates to find the limit at the infinity for this ratio. Derived estimates are the generalization of the well-known result in Riemannian geometry. We also estimate the v...
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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
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This paper investigates the metric structure of compact -parabolic surfaces and topological properties of -saddle surfaces in the sense of Šefel' in symmetric spaces of rank one, namely, spherical space , complex projective space , and quaternion projective space . It turns out that -parabolic surfaces for large are totally geodesic spheres in , to...
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This paper investigates surfaces of nonpositive extrinsic curvature in a pseudo- Riemannian space of curvature 1, Kählerian submanifolds of complex projective space , and saddle surfaces in spherical space . It is determined under what conditions a surface is a totally geodesic submanifold.Bibliography: 14 titles.
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Locally convex compact immersed hypersurfaces in the Finsler—Hadamard space with bounded T-curvature are considered. Under certain conditions on normal curvatures, such hypersurfaces are proved to be convex, embedded, and homeomorphic to the sphere. To this end, the Rauch theorem is generalized to exponential maps of hypersurfaces and the convexity...
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We prove a theorem about an extremal property of Lobachevsky space among simply connected Riemannian manifolds of nonpositive curvature.
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We introduce an analog of the Chern-Lashof absolute curvature for complex submanifolds in complex Euclidean spaces. A relation between this curvature and the volume of the Grassmann image of the submanifold is established.
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The (k,ε)-saddle (in particular, k-saddle, i.e. ε=0) submanifolds are defined in terms of eigenvalues of the second fundamental form. This class extends the class of submanifolds with extrinsic curvature bounded from above, i.e. ⩽ε2 (in particular, non-positive) and small codimension. We study s-connectedness and (co)homology properties of compact...
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The paper is directed to lay people in the main subject of it (Ricci flow) and also in the topological background of the problem. Then, the general philosophy of these lectures is to begin with the more elementary facts, give some details on them (sometimes many details), and introduce to the more advanced topics, with a decreasing exposition of de...
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We state several sufficient conditions for compact spacelike surface in the3-dimensional de Sitter space to be totally geodesic or spherical.
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We give sharp upper estimates for the difference circumradius minus inradius and for the angle between the radial vector (respect to the center of an inball) and the normal to the boundary of a compact h,λ-convex domain in the Hadamard manifold. We apply these estimates to get the limit at the infinity for the quotients Volume/Area and (Total k-mea...
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In a Hadamard manifold with sectional curvaturebounded from below by –k 2 2, we give sharp upper estimates for the difference circumradius minus inradiusof a compact k 2-convex domain, and we getalso estimates for the quotient (Total d-mean curvature)/Area of a convex domain.
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In a Hadamard manifold with sectional curvature bounded from below by −k 2 2 , we give sharp upper estimates for the difference circumradius minus inradius of a compact k 2-convex domain, and we get also estimates for the quotient (Total d-mean curvature)/Area of a convex domain. Mathematics Subject Classifications (2000): 53C40, 53C42, 53C21, 53C4...
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This survey contains results on local and global isometric immersions of two-dimensional and multidimensional Riemannian and pseudo-Riemannian space forms into spaces of constant sectional curvature. Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Abstract Text Return: Q...
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It is known that for a sequence of convex sets expanding over the whole hyperbolic space the limit of the quotient is less or equal than 1/n, and exactly 1/n when the sets considered are convex with respect to horocycles. When convexity is with respect to equidistant lines, i.e., curves with constant geodesic curvature λ less than one, the above li...
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It is known that a tube over a Kähler submanifold in a complex space form is a Hopf hypersurface. In some sense the reverse statement is true: a connected compact generic immersed C2n-1 regular Hopf hypersurface in the complex projective space is a tube over an irreducible algebraic variety. In the complex hyperbolic space a connected compact gener...
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Locally convex compact hypersurfaces immersed in a hollow simply connected Riemannian space of nonpositive sectional curvature are considered. They are proved to be convex hypersurfaces homeomorphic to the sphere. A similar result for immersed hypersurfaces with nonpositive definite second quadratic form of rank no smaller than one is obtained.
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It is shown that the structure of a three-dimensional minimal parabolic surface is determined by the pair (V2, γ), where V2 is a minimal two-dimensional surface in Sn and γ satisfies Δγ+2γ=0 (here Δ is the Laplace operator in ℝ4). It is also shown that the singularities of the surface are determined by zeros of γ. Bibliography: 9 titles.
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We give sharp upper estimates for the difference circumradius minus inradius and for the angle between the radial vector (respect to the center of an inball) and the normal to the boundary of a compact $h$-convex domain in the hyperpolic space. We apply these estimates to get the limit at the infinity for the quotients Volume/Area and (Total $k$-me...
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Not Available Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints
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ContentsIntroduction § 1. Parabolic submanifolds1.1. Point structure of the space of second fundamental forms of a k-parabolic surface1.2. Normal bundle of a manifold1.3. Local structure of the k-parabolic submanifolds in a Riemannian space1.4. Complete parabolic submanifolds of a Riemannian space1.5. Pontryagin characteristic classes of parabolic...
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Contents §1. Introduction §2. Affine classification of points of multidimensional surfaces2.1. Main results2.2. Proof of Theorem 2.1.12.3. Classification of points of 3-dimensional submanifolds §3. Strongly parabolic submanifolds3.1. Local structure of a strongly parabolic surface in Riemannian space3.2. Local structure of a strongly parabolic surf...
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The topological structure of compact Riemannian manifolds that admit hyperbolic foliations is studied.
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Saddle submanifolds are considered. A characterization of such submanifolds of Euclidean space is given in terms of sectional curvature. Extending results of T. Frankel, K. Kenmotsu and C. Xia, we determine under what conditions two complete saddle submanifolds of a complete connected Riemannian manifold M, with nonnegative k-Ricci curvature, must...
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In this paper we consider compact multidimensional surfaces of nonpositive external curvature in a Riemannian space. If the curvature of the underlying space is ≥ 1 and the curvature of the surface is ≤ 1, then in small codimension the surface is a totally geodesic submanifold that is locally isometric to the sphere. Under stricter restrictions on...
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The following theorem is proved. THEOREM. If on an infinite, complete, convex hypersurface F in E4 the mean curvature is 1 − ε ≤ H ≤ 1, where 0 ≤ ε ≤ 10−11, then F is a cylindrical hypersurface.
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CONTENTS Introduction § 1. Riemannian and pseudo-Riemannian metrics on fibre bundles § 2. Riemannian submersions 2.1. Main equations of a Riemannian submersion 2.2. Geodesics in Riemannian submersions § 3. Connection between the geometric features of the tangent bundle (the normal bundle) and the base § 4. Geodesic lines in the tangent and normal b...
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CONTENTS I. Introduction II. Topology of Grassmann manifolds 1. Local coordinates 2. The cell decomposition and basic topological characteristics 3. Plücker coordinates III. Riemannian geometry of Grassmann manifolds: geometric approach 1. The metric and angles between planes 2. Curvature tensor, sectional curvature, closed geodesics, the limit set...
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We study complete, strongly parabolic metrics with constant relative index of nullity μc=k and complete, strongly parabolic surfaces with constant index of relative nullity v=k in a constant curvature space Rn(c) under the assumption that there exists a surface orthogonal to the fibers of total geodesity, and if c<0, under the additional condition...
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It is known that a k-strongly parabolic surface in Euclidean space (index of relative nullity is at least k) inherits a k-strongly parabolic metric (index of nullity is at least k). For the definitions of the index of relative nullity ?(x) and the index of nullity µ(x) see Sh. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscie...
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Sufficient conditions are found for complete parabolic surfaces to be cylindrical in Euclidean space. A complete (l-2)-[resp. (l-3)]-parabolic surface is a cylinder with (l-3)-[resp. (l-4)]-dimensional generators if the order of flattening of the surface isl-2 [resp.l-3] and the strong zero-index is constant.
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We study the tangent bundle of vectors of fixed length on a Riemannian manifold. We give sufficient conditions for the sectional curvature of the Sasaki metric on the tangent bundle of vectors of fixed length to be nonnegative.

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