arXiv research

The study defines and characterizes extrinsic catenaries in hyperbolic space.

Catenaries defined on any Riemannian surface using intrinsic distance.

The paper studies dual catenaries in the dual plane, deriving equations and characterizations.

The paper defines catenary curves in spheres and hyperbolic planes.

Paper proves uniqueness of catenary cylinders based on their asymptotic shape.

The paper studies hanging chains and surfaces in degenerate geometries.

In this paper we generalize in Lorentz-Minkowski space $ł^3$ the two-dimensional analogue of the catenary of Euclidean space. We solve the Dirichlet problem for bounded mean convex domains and spacelike boundary data that have a spacelike extension to the domain. We also classify all singular maximal surfaces of $ł^3$ …

The paper proves cylindrical nature of singular minimal ruled surfaces.

The paper explores singular minimal translation graphs in Euclidean spaces.

We consider an appoximation of a catenoid constructed from "odd" truncated cones that maintains minimality in a certain sense. Thorough this procedure, we obtain a discrete curve approximating a catenary by exploiting the fact that it is the function that generates a catenoid. In this investigation, the theory of the G…

Explicit formulas for extrinsic Paneitz operators and Q-curvatures for totally umbilic hypersurfaces.

Characterizes curves for minimal surfaces in de Sitter space.

The extrinsic Bonnet-Myers theorem is proven for positive Ricci curvature manifolds.

We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kaehler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms.

We find a one-to-one correspondence between full extrinsic symmetric spaces in (possibly degenerate) inner product spaces and certain algebraic objects called (weak) extrinsic symmetric triples. In particular, this yields a description of arbitrary extrinsic symmetric spaces in pseudo-Euclidean spaces by corresponding …

Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group of an indefinite extrinsic symmetric space is not semisimple, which makes the cl…

Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.

New operators and curvatures derived from embedded manifolds.

Analyzes canonical reductive decomposition of extrinsic homogeneous submanifolds.

In this article we introduce a generalization of the Newton transformation to the case of a system of endomorphisms. We show that it can be used in the context of extrinsic geometry of foliations and distributions yielding new integral formulas containing generalized extrinsic curvatures.

Novel coarse extrinsic curvature for Riemannian submanifolds.

We study infinitesimal semi-simple extrinsic symmetric spaces and give a classification in the symplectic case.

We study the topology of (properly) immersed complete minimal surfaces $P^2$ in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces, (see \cite{Pa}). We present an alternative and partially unified proof of…

We prove the energy identity and the no neck property for a sequence of smooth extrinsic polyharmonic maps with bounded total energy.

Extrinsic Geometric Flow (EGF) for a codimension-one foliation has been recently introduced by authors as deformations of Riemannian metrics subject to quantities expressed in terms of its second fundamental form. In the paper we introduce soliton solutions to EGF and study their geometry for totally umbilical foliatio…

In the paper we prove, that extrinsic curvature does not impose restrictions on the topology of a contact structure, except the obvious ones.

The diameter of a disc filling a loop in the universal covering of a Riemannian manifold may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram filling a word that represents the i…

Classifies minimal submanifolds in complex hyperbolic spaces.

In this note, we prove lower and upper bounds for Dirac operators of submanifolds in certain ambient manifolds in terms of conformal and extrinsic quantities.

We prove that complete submanifolds, on which the Omori-Yau weak maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the la…

The following Theorem is proved: Let M be an n-dimensional (n>2) submanifold of a Riemannian manifold N. Suppose that through each point p of M there exist two (n-1)-dimensional extrinsic spheres of N, which are contained in M in a neighbourhood of p and are tangent to each other at p. Then M is totally geodesic in N o…

Discretizations of the mean curvature and extrinsic curvature components are constructed on piecewise flat simplicial manifolds, giving approximations for smooth curvature values in a mostly mesh-independent way. These constructions are given in combinatoric form in terms of the extrinsic hinge angles, the intrinsic st…

We obtain an optimal estimate for the extrinsic curvature of an entire minimal graph in $\H^2\times\R$, $\H^2$ the hyperbolic plane.

Derives formulas for extrinsic Paneitz operator and $Q$-curvature in general dimensions.

Proves uniqueness of geometric flow in various Riemannian manifolds.

We study, from the extrinsic point of view, the structure at infinity of open submanifolds isometrically immersed in the real space forms of constant sectional curvature $κ\leq 0$. We shall use the decay of the second fundamental form of the the so-called tamed immersions to obtain a description at infinity of the subm…

Paper proves a Penrose inequality in extrinsic geometry.

Paper finds convexity in translating solitons for concave flows.

Study proves inequality for hypersurfaces and shows almost extremals are close to Wulff shape.

Improving a result of Eschenburg and Kim we give a criterion for semisimplicity of pseudo-Riemannian extrinsic symmetric spaces in terms of the shape operator with respect to the mean curvature vector.

Geodesic spheres are the only quasicomplete surfaces in 3-space-forms.

Researchers define residue families and use them to solve singular Yamabe problems.

In this article, we give the integrability conditions for the existence of an isometric immersion from an orientable simply connected surface having prescribed Gauss map and positive extrinsic curvature into some unimodular Lie groups. In particular, we discuss the case when the Lie group is the euclidean unit sphere $…

Unified study of surfaces using Clifford algebras.

Study on biconservative surfaces in 4D hyperbolic space, providing extrinsic descriptions.

New systems of linear PDEs discovered in 3D contact manifolds.

We study deformations of Riemannian metrics on a given manifold equipped with a codimension-one foliation subject to quantities expressed in terms of its second fundamental form. We prove the local existence and uniqueness theorem and estimate the existence time of solutions for some particular cases. The key step of t…

The study characterizes constant curvature manifolds using ruled surfaces.