arXiv research

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

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

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…

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…

We study the problem of isometrically embedding a two-dimensional Riemannian manifold into Euclidean three-space. It is shown that if Gaussian curvature vanishes to finite order and its zero set consists of two smooth curves tangent at a point, then local sufficiently smooth isometric embedding exists.

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…

Cylinders in warped product spaces have zero curvature.

Study modular surfaces in Lorentz-Minkowski 3-space, classifying and analyzing their curvature and applications.

We study the canonical metric on a compact Riemann surface of genus at least two. While it is known that the canonical metric is of nonpositive curvature, we show that its Gaussian curvatures are not bounded away from zero nor negative infinity when the surface is close to the compactification divisor of Riemann's modu…

We study the old problem of isometrically embedding a 2-dimensional Riemannian manifold into Euclidean 3-space. It is shown that if the Gaussian curvature vanishes to finite order and its zero set consists of two Lipschitz curves intersecting transversely at a point, then local sufficiently smooth isometric embeddings …

Study Hamiltonian stationary Lagrangian surfaces in complex space forms.

In this paper we study surfaces foliated by a uniparametric family of circles in the homogeneous space Sol$_3$. We prove that there do not exist such surfaces with zero mean curvature or with zero Gaussian curvature. We extend this study considering surfaces foliated by geodesics, equidistant lines or horocycles in tot…

Paper proves rigidity for self-similar solutions in 3D flows.

In this paper, we generalize our results in \cite{GX3} to triangulated surfaces in hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gaussian curvature by dividing the classical discrete Gauss curvature by an area element, which …

The paper finds surfaces closest to being flat that span a given contour.

In a previous paper we classified complete stationary surfaces (i.e. spacelike surfaces with zero mean curvature) in 4-dimensional Lorentz space $\mathbb{R}^4_1$ which are algebraic and with total Gaussian curvature $-\int K\mathrm{d}M=4π$. Here we go on with the study of such surfaces with $-\int K\mathrm{d}M=6π$. It …

Third-order PDEs describe spherical and pseudospherical surfaces.

Formulae quantify gaps in geodesic quadrilaterals on manifolds.

This paper extends, in a sharp way, the famous Efimov's Theorem to immersed ends in $\real^3$. More precisely, let $M$ be a non-compact connected surface with compact boundary. Then there is no complete isometric immersion of $M$ into $\Bbb R^3$ satisfying that $\int_M |K|=+\infty$ and $K\le-κ<0$, where $κ$ is a positi…

The paper classifies equations describing spherical or pseudospherical surfaces.

Study discrete analog of zeta-determinant maximization on triangulated surfaces.

Applying the general theory about complete spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space $\mathbb{R}^4_1$, we classify those regular algebraic ones with total Gaussian curvature $-\int K\mathrm{d}M=4π$. Such surfaces must be oriented and be congruent to either the generalized c…

Study on surfaces with constant anisotropic mean curvature in 3D space.

In this paper we define and analyze singularities of discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms. In particular, we discuss singularities of discrete surfaces with non-zero constant Gaussian curvature, and parallel surfaces of discret…

The aim of this paper is to associate a measure for certain sets of paths in the Euclidean plane $\mathbb{R}^2$ with fixed starting and ending points. Then, working on parameterized surfaces with a specific Riemannian metric, we define and calculate the integral of the length over the set of paths obtained as the image…

Study sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces.

We determine the expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective…

The paper classifies translation surfaces with constant curvature in a specific connection.

A surface is called a tube if its level-sets with respect to some coordinate function (the axis of the surface) are compact. Any tube of zero mean curvature has an invariant, the so-called flow vector. We study how the geometry of the Gaussian image of a higher-dimensional minimal tube M is controlled by the angle alph…

We show that the results in \cite{Ge-Jiang1} are still true in hyperbolic background geometry setting, that is, the solution to Chow-Luo's combinatorial Ricci flow can always be extended to a solution that exists for all time, furthermore, the extended solution converges exponentially fast if and only if there exists a…

Zero-inflated datasets, which have an excess of zero outputs, are commonly encountered in problems such as climate or rare event modelling. Conventional machine learning approaches tend to overestimate the non-zeros leading to poor performance. We propose a novel model family of zero-inflated Gaussian processes (ZiGP) …

We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces, and then classify them. We then define and analyze their singularities. In partic…

We classify (spacelike or timelike) surfaces of revolution with zero $f$-mean curvature in $\Bbb G^2\times\Bbb R_1,$ the Lorentz-Minkowski 3-space $\Bbb R^3_1$ endowed with the Gaussian-Euclidean density $e^{-f(x,y,z)}=\frac 1{2π}e^{-\frac{x^2+y^2}2}.$ It is proved that an $f$-maximal surface of revolution is either a …

Study on zeros of Gaussian sections on semipositive line bundles on punctured Riemann surfaces.

Paper solves curvature prescription problem on surfaces with boundary.

Polyhedra can mimic constant curvature surfaces, even with self-intersections.

Study complete gradient Ricci solitons with zero radial Weyl curvature.

Study classifies zero mean curvature surfaces with planar curvature lines.

The paper derives new Gauss-Bonnet formulas for frontal bundles over surfaces with boundary.

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.

The study proves the finiteness of moments for Gaussian field zeros and critical points.

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.

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…