arXiv research

Examines discrete curvature's relation to smooth curvature in 3 spaces.

Paper defines curvature equivalence for Legendre curves in a plane.

We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…

A diffeomorphism of pseudo-Riemannian manifolds is called sectional curvature preserving if it preserves the sectional curvature of all the nondegenerate 2-planes. We consider a similar condition for degenerate 2-planes and we prove that the diffeomorphism is conformal (when the condition is fulfilled for weakly degene…

The paper examines asymptotic lines of plane fields in 3D space.

Study metrics on half plane with specific curvature properties.

The study finds bounds on metrics with constant curvature in the plane.

Computed distortion coefficients for the α-Grushin plane.

Study on plane curves with special connections and curvatures.

In this paper results from the differential geometry of curves are extended from normed planes to gauge planes which are obtained by neglecting the symmetry axiom. Based on the gauge analogue of the notion of Birkhoff orthogonality from Banach space theory, we study all curvature types of curves in gauge planes, thus g…

New example of surface flow converging to a plane with multiplicity 2.

Study variational properties of curves in half-plane with area constraints.

We classify all real hypersurfaces with constant principal curvatures in the complex hyperbolic plane.

We study rays in von Mangoldt planes, which has applications to the structure of open complete manifolds with lower radial curvature bounds. We prove that the set of souls of any rotationally symmetric plane of nonnegative curvature is a closed ball, and if the plane is von Mangoldt, we compute the radius of the ball. …

Analytic plane curves determine unique conformal coordinates.

Study curvatures of diffeomorphisms on non-orientable surfaces.

Study on real hypersurfaces in complex projective plane with constant mean curvature.

Research explores Lorentzian distances on a specific geometric plane.

The paper calculates curvature limits and proves Gauss-Bonnet theorems in affine and Minkowski groups.

It is well known that plane curves with the same endpoints are homotopic. An analogous claim for plane curves with the same endpoints and bounded curvature still remains open. In this work we find necessary and sufficient conditions for two plane curves with bounded curvature to be deformed, one to another, by a contin…

The Funk-Finsler structure is constructed in hyperbolic models, including the Klein unit disc.

Suppose curves are moving by curvature in a plane, but one embeds the plane in $R^3$ and looks at the plane from an angle. Then circles shrinking to a round point would appear to be ellipses shrinking to an ``elliptical point,'' and the surface energy would appear to be anisotropic as would the mobility. The result of …

The paper classifies curves in dual affine and Lorentz-Minkowski planes with constant curvature.

We determine non-Hopf hypersurfaces with constant mean curvature in the complex projective plane which attain equality in a basic inequality between the maximum Ricci curvature and the squared mean curvature.

New sub-Riemannian spaces with boundary meet curvature-dimension condition.

New Ricci curvature means derived from plane curvatures.

We explicitly determine tori that have a parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane

Proves conjecture about geodesic foliations in Riemannian planes.

Riemann zero mean curvature examples in the Lorentz-Minkowski space are surfaces with zero mean curvature foliated by circles contained in parallel planes. In contrast to the Euclidean case, this family of surfaces presents new and rich features because of the variety of types of circles. In this paper, we give a geome…

We prove that S^2 x S^2 satisfies an intermediate condition between having metrics with positive Ricci and positive sectional curvature. Namely, there exist metrics for which the average of the sectional curvatures of any two planes tangent at the same point, but separated by a minimum distance in the 2-Grassmannian, i…

Veronese minimizes normal curvatures to sphere.

We classify all Hamiltonian stationary Lagrangian surfaces in complex Euclidean plane which are self-similar solutions of the mean curvature flow.

We prove that the Euclidean plane is the only Riemannian plane with total curvature and free of conjugate points that satisfies Playfair's version of the parallel postulate.

A discrete method approximates hyperbolic curvature flow in the plane.

We prove that a closed immersed plane curve with total curvature $2πm$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature $2πm$ whose entropy is less than $m$ …

Study a flow preserving area of plane curves, ending in a circle.

The study glues $CAT(0)$ subsets of the plane under certain curvature conditions.

Using certain solutions of the curve shortening flow, including self-shrinking and self-expanding curves or spirals, we construct and characterize many new examples of translating solitons for mean curvature flow in complex Euclidean plane. They generalize the Joyce, Lee and Tsui ones \cite{JLT} in dimension two. The s…

Study on closed $p$-elastic curves in hyperbolic and de Sitter planes.

In this paper we obtain two types of optimal inequalities consisting of the normalized scalar curvature and the generalized normalized $δ$-Casorati curvatures for real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians. We also find the conditions on which the equalities hol…

The paper proves existence and behavior of Lagrangian tori in complex projective plane.

We calculate the Riemann curvature tensor and sectional curvature for the Lie group of volume-preserving diffeomorphisms of the Klein bottle and projective plane. In particular, we investigate the sign of the sectional curvature, and find a possible disagreement with a theorem of Lukatskii. We suggest an amendment to t…

The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curva…

Ricci flow deforms metrics with positive curvature to include negative curvature.

Study of closed real plane curves with hyperelliptic genus three solutions.

Paper solves dual Minkowski problem in 2D plane for specific curvature cases.

We find the first examples of real hypersurfaces with two nonconstant principal curvatures in complex projective and hyperbolic planes, and we classify them. It turns out that each such hypersurface is foliated by equidistant Lagrangian flat surfaces with parallel mean curvature or, equivalently, by principal orbits of…

The aim of this work is studying translating graphs by mean curvature flow in $\Real^3$. We prove non-existence of complete translating graphs over bounded domains in $\Real^2$. Furthermore, we show that there are only three types of complete translating graphs in $\Real^3$; entire graphs, graphs between two vertical p…