arXiv research

This article is an application of the author's paper about a construction method for discrete constant negative Gaussian curvature surfaces, the nonlinear d'Alembert formula. The heart of this formula is the Birkhoff decomposition, and we give a simple algorithm for the Birkhoff decomposition. As an application, we dra…

New Bäcklund transformations for discrete pseudospherical surfaces of revolution are found.

The study investigates deformations of swallowtails in 3D space, preserving curvature signs.

Let $M$ be a closed oriented surface of negative Gaussian curvature and let $Ω$ be a non-exact 2-form. Let $λ$ be a small positive real number. We show that the longitudinal KAM-cocycle of the magnetic flow given by $\la Ω$ is a coboundary if and only if the Gaussian curvature is constant and $Ω$ is a constant multiple…

The problem of prescribing Gaussian curvature on Riemann surface with conical singularity is considered. Let $(Σ,β)$ be a closed Riemann surface with a divisor $β$, and $K_λ=K+λ$, where $K:Σ\rightarrow\mathbb{R}$ is a Hölder continuous function satisfying $\max_ΣK= 0$, $K\not\equiv 0$, and $λ\in\mathbb{R}$. If the Eule…

The article proves Randers Poincaré disc satisfies isoperimetric equality.

Given an smooth function $K <0$ we prove a result by Berger, Kazhdan and others that in every conformal class there exists a metric which attains this function as its Gaussian curvature for a compact Riemann surface of genus $g>1$. We do so by minimizing an appropriate functional using elementary analysis. In particula…

Complete Finsler spaces with negative Ricci curvature are reversible.

New metrics found without topological restrictions.

Integrable nets described with curvature relations to pseudospherical surfaces.

The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik…

Constructs metrics with negative constant scalar curvature.

In this paper we study constant positive Gauss curvature $K$ surfaces in the 3-sphere $S^3$ with $0<K<1$ as well as constant negative curvature surfaces. We show that the so-called normal Gauss map for a surface in $S^3$ with Gauss curvature $K<1$ is Lorentz harmonic with respect to the metric induced by the second fun…

Conditions ensure constant curvature in negatively curved manifolds.

The paper solves curvature prescription problems on balls and disks.

Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.

Sharp Sobolev inequalities proved on manifolds with non-negative Ricci curvature.

The paper solves curvature prescription on a disk with negative Gaussian curvature.

Study negative scalar curvature metrics with positive boundary mean curvature.

Negative curvature manifolds have vanishing bounded volume class if and only if Cheeger constant is positive.

We consider operators $L$ acting on functions on a Riemannian surface, $Σ$, of the form $L = Δ+ V +a K.$ Here $Δ$ is the Laplacian of $Σ$, $V$ a non-negative potential on $Σ$, K the Gaussian curvature and $a$ is a non-negative constant. Such operators $L$ arise as the stability operator of $Σ$ immersed in a Riemannian …

Directly proves Li-Yau estimates on manifolds with negative Ricci curvature.

Study of Ricci flow on discrete surfaces of revolution with constant Gaussian curvature.

The study classifies warped products with harmonic curvature on surfaces, showing two possibilities for the metric.

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…

Let $g$ be a metric on the $2$-sphere $\mathbb{S}^2$ with positive Gaussian curvature and $H$ be a positive constant. Under suitable conditions on $(g, H)$, we construct smooth, asymptotically flat $3$-manifolds $M$ with non-negative scalar curvature, with outer-minimizing boundary isometric to $(\mathbb{S}^2, g)$ and …

The study proves inequalities and curvature properties for Markov chains.

In this paper we classify compact minimal surfaces in $S^5$ with non-negative Gaussian curvature using the notion of a contact angle.

Study geodesic curvature of logarithmic spirals on curved surfaces.

Authors construct hypertori with constant negative mean curvature in a sphere.

The paper proves isoperimetric inequalities in manifolds with small negative Ricci curvature.

Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.

Classifies surfaces in hyperbolic space with constant Gaussian curvature.

The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finsler metrics on an open subset in R^n with negative flag curvature and constant S-curvature. In th…

Wave maps (or Lorentzian-harmonic maps) from a $1+1$-dimensional Lorentz space into the $2$-sphere are associated to constant negative Gaussian curvature surfaces in Euclidean 3-space via the Gauss map, which is harmonic with respect to the metric induced by the second fundamental form. We give a method for constructin…

We investigate geometric aspects of the the Bäcklund transform of principal contact element nets. A Bäcklund transform exists if and only if it the principal contact element net is of constant negative Gaussian curvature (a pseudosphere). We describe an elementary construction of the Bäcklund transform and prove its co…

The paper solves a problem in metric geometry for disks with negative curvature.

Flat surfaces in Lie groups with constant curvature are flat.

We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Thus we extend well known notions of discrete pseudospherical surfaces and smooth pseudosperical surfaces on more exotic domains (e.g, the Cantor set). In particular, we present a new expression for the discrete Gauss…

We consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a Riemannian metric $g$ of negative curvature with fixed total area. The second author has shown that the topological entropy of geodesic flow for $g$ is greater than or equal to the topological entropy for the metric of constant negative curvatu…

The paper classifies metrics with constant negative Q-curvature in Euclidean spaces.

The study proves constant-curvature analogues of hot spots conjecture for triangles.

The study proves Lieb-Thirring inequalities on hyperbolic manifolds.

In this paper, we show that the constant property of the Gaussian curvature of surfaces of revolution in both $\mathbb R^4$ and $\mathbb R_1^4$ depend only on the radius of rotation. We then give necessary and sufficient conditions for the Gaussian curvature of the general rotational surfaces whose meridians lie in two…

For negatively curved manifolds, a condition is found for intrinsic ultracontractivity of heat semigroups.

It is proved that if an AK2-manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then it is a 6-dimensional manifold of constant negative sectional curvature or a Kähler manifold of constant holomorphic sectional curvature.

We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete Riemannian manifolds of constant negative scalar curvature are isometries. We als…

Hilbert-Efimov theorem states that any complete surface with curvature bounded above by a negative constant can not be isometrically imbedded in $\mathbb{R}^3.$ We demonstrate that any simply-connected smooth complete surface with curvature bounded above by a negative constant admits a smooth isometric embedding into t…