arXiv research

The paper proves conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.

Study complete gradient Ricci solitons with zero radial Weyl curvature.

Study classifies zero mean curvature surfaces with planar curvature lines.

Study duality of zero mean curvature surfaces in Heisenberg group.

Classifies surfaces with zero mean curvature in a light cone.

Solves surface problem in 3D light cone.

New examples of mixed-type zero-curvature graphs found.

On any timelike surface with zero mean curvature in the four-dimensional Minkowski space we introduce special geometric (canonical) parameters and prove that the Gauss curvature and the normal curvature of the surface satisfy a system of two natural partial differential equations. Conversely, any two solutions to this …

Paper investigates reflection principles for zero mean curvature surfaces in isotropic 3-space.

In this paper we present a local description for complete minimal hypersurfaces in $\mathbb{S}^5$ with zero Gauss-Kronecker curvature, zero $3$-mean curvature and nowhere zero second fundamental form.

The class of second order ODE's cubic with respect to the first order derivative is considered. Using geometric structures associated with these equations, the subclasses of umbilical equations, zero mean curvature equations, and zero Gaussian curvature equations are defined. Zero mean curvature equations are studied w…

In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bo…

Study on surfaces in neutral space forms with zero mean curvature.

The paper connects bundle curvature to random zero currents.

Study on $ρ$-Einstein solitons with zero scalar curvature, proving stability and flatness.

New framework for zero mean curvature surfaces in isotropic 3-space.

The paper examines translating solitons and their relation to Lagrangian mean curvature flows with zero Maslov class.

Minimal hypersurfaces in S^5 with specific curvature properties are totally geodesic.

A general formulation of zero curvature connections in a principle bundle is presented and some applications are discussed. It is proved that a related connection based on a prolongation in an associated bundle remains zero curvature as well. It is also shown that the connection coefficients can be defined so that the …

We investigate complete minimal hypersurfaces in the Euclidean space $$, with Gauss-Kronecker curvature identically zero. We prove that, if $f:M^{3}\to {R}^{4}$ is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature b…

A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and as a corollary, we prove that if …

In Finsler geometry, there are infinitely many models of constant curvature. The Funk metrics, the Hilbert-Klein metrics and the Bryant metrics are projectively flat with non-zero constant curvature. A recent example constructed by the author is projectively flat with zero curvature. In this paper, we introduce a techn…

Study finds Scherk type surfaces as extremals for zero-curvature minimal graphs.

The paper solves conditions for prescribing scalar and Gauss curvatures on manifolds with zero first eigenvalue.

It is classically known that the only zero mean curvature entire graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ is called of mixed type if it changes causal type from space-like to time-like. In $\boldsymbol{R}^3_1$, Osamu Kobayashi found …

Paper proves Chern flat for 3D Hermitian manifolds with zero real bisectional curvature.

Study shows simplicial volume of certain fiber bundles is zero.

Factorable surfaces, i.e. graphs associated with the product of two functions of one variable, constitute a wide class of surfaces. Such surfaces in the pseudo-Galilean space with zero Gaussian and mean curvature were obtained in [1]. In this study, we provide new classification results relating to the factorable surfa…

No regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces are found.

We investigate the structure of 3-dimensional complete minimal hypersurfaces in the unit sphere with Gauss-Kronecker curvature identically zero.

The method of contact integrable extensions is used to find new zero-curvature representation for Plebañski's second heavenly equation.

Classifies zero mean curvature surfaces in Lorentz-Minkowski space.

The paper proves properties of Kähler surfaces with zero scalar curvature.

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…

Statistical Lie algebras with constant curvature are linked to locally conformally Kähler structures.

Space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski3-space are both characterized as zero mean curvature surfaces. We are interested in the case where the zero mean curvature surface changes type from space-like to time-like at a given non-degenerate null curve. We consider this phenomenon a…

Study rigidity of Ricci flow limits on nilpotent bundles with zero curvature.

Defines a metric and form for a bundle moduli space, leading to a zero-curvature formulation.

New, algebraic surfaces found in curved spaces.

We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space $\mathbb{H}^{4}$ with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and …

Cylinders in warped product spaces have zero curvature.

Unbounded convex domains have zero mean curvature on disconnected boundaries.

The paper shows how Scherk-type surfaces can be decomposed into helicoids.

In this work, we study a class of rotational surfaces in the pseudo-Euclidean space $\mathbb{E}_2^4$ whose profile curves lie in two-dimensional planes. We solve the differential equation that characterizes the rotational surfaces with zero mean curvature to determine the profile curves of such rotational surfaces. The…

This paper studies global webs on the projective plane with vanishing curvature. The study is based on an interplay of local and global arguments. The main local ingredient is a criterium for the regularity of the curvature at the neighborhood of a generic point of the discriminant. The main global ingredient, the Lege…

We will prove that \emph{there are no stable complete hypersurfaces of $\mathbb{R}^4$ with zero scalar curvature, polynomial volume growth and such that $\dfrac{(-K)}{H^3}\geq c>0$ everywhere, for some constant $c>0$}, where $K$ denotes the Gauss-Kronecker curvature and $H$ denotes the mean curvature of the immersion. …

We construct triply periodic zero mean curvature surfaces of mixed type in the Lorentz-Minkowski 3-space, with the same topology as the triply periodic minimal surfaces in the Euclidean 3-space, called Schwarz rPD surfaces.

We construct embedded triply periodic zero mean curvature surfaces of mixed type in the Lorentz-Minkowski 3-space with the same topology as the Schwarz D surface in the Euclidean 3-space.