arXiv research

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

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…

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

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

The paper examines conditions for Lagrangian surfaces in Kähler-Einstein manifolds.

Let $h$ be a complete metric of Gaussian curvature $K_0$ on a punctured Riemann surface of genus $g \geq 1$ (or the sphere with at least three punctures). Given a smooth negative function $K$ with $K=K_0$ in neighbourhoods of the punctures we prove that there exists a metric conformal to $h$ which attains this function…

In this paper, we study the power of Gaussian curvature flow of a compact convex hypersurface and establish its Harnack inequality when the power is negative. In the Harnack inequality, we require that the absolute value of the power is strictly positive and strictly less than the inverse of the dimension of the hypers…

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…

Study relates Gaussian curvature signs to cuspidal edge types and geometric invariants.

The paper studies ray transforms on surfaces with negative curvature, proving injectivity and determining connections and Higgs fields.

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

We classify complete biharmonic surfaces with parallel mean curvature vector field and non-negative Gaussian curvature in complex space forms.

We use a Simons type equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.

The torus cannot collapse to a segment under certain curvature conditions.

In this note, we study the curvature flow to Nirenberg problem on $S^2$ with non-negative nonlinearity. This flow was introduced by Brendle and Struwe. Our result is that the Nirenberg problems has a solution provided the prescribed non-negative Gaussian curvature $f$ has its positive part, which possesses non-degenera…

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…

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…

New heat kernel bounds on manifolds with non-negative Ricci curvature.

Study on compact Kähler surfaces for sign-changing curvatures.

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…

Study on surfaces with conical singularities and geodesic boundaries, deriving existence results.

We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is an $n$-dimensional space form. Then, we use this equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.

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

Integrable nets described with curvature relations to pseudospherical surfaces.

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

The study proves inequalities and curvature properties for Markov chains.

The article proves Randers Poincaré disc satisfies isoperimetric equality.

Extends heat kernel estimates for super Ricci flow.

We consider hypersurfaces in the real Euclidean space $\mathbb{R}^{n+1}$ ($n\geq2$) which are relatively normalized. We give necessary and sufficient conditions a) for a surface of negative Gaussian curvature in $\mathbb{R}^3$ to be ruled, b) for a hypersurface of positive Gaussian curvature in $\mathbb{R}^{n+1}$ to be…

The study bounds heat kernel for manifolds with specific curvature conditions.

Chirality affects the curvature of molecular networks, influencing their shape and stability.

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

For any closed Riemannian manifold $X$ we prove that large isoperimetric regions in $X\times{\mathbb R}^n$ are of the form $X\times$(Euclidean ball). We prove that if $X$ has non-negative Ricci curvature then the only soap bubbles enclosing a large volume are the products $X\times$(Euclidean sphere). We give an example…

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…

New proof shows special surfaces have finite type.

This article shows that under locally uniformly integral bounds of the negative part of Ricci curvature the heat kernel admits a Gaussian upper bound for small times. This provides general assumptions on the geometry of a manifold such that certain function spaces are in the Kato class. Additionally, the results imply …

In \cite{ly, ly2}, Liu and the second author propose a definition of the quasi-local mass and prove its positivity. This is demonstrated through an inequality which in turn can be interpreted as a total mean curvature comparison theorem for isometric embeddings of a surface of positive Gaussian curvature. The Riemannia…

In this work, we study graphs in $\M^n\times\Real$ that are evolving by the mean curvature flow over a bounded domain on $\M^n$, with prescribed contact angle in the boundary. We prove that solutions converge to translating surfaces in $\M^n\times\Real$. Also, for a Riemannian manifold $\M^2$ with negative Gaussian cur…

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

We study the isoperimetric, functional and concentration properties of $n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $N$ is negative, and more generally, is in the range $N \in (-\infty,1)$, extending the scope from the traditional range $N \i…

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…

The paper solves curvature prescription problems on balls and disks.

In this paper, we study curvature dimension conditions on birth-death processes which correspond to linear graphs, i.e., weighted graphs supported on the infinite line or the half line. We give a combinatorial characterization of Bakry and Émery's $CD(K,n)$ condition for linear graphs and prove the triviality of edge w…

We define the notion of special Lagrangian curvature, showing how it may be interpreted as an alternative higher dimensional generalisation of two dimensional Gaussian curvature. We obtain first a local rigidity result for this curvature when the ambiant manifold has negative sectional curvature. We then show how this …

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…

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

Non-negative $L_1$-approximating polynomials for Gaussian distributions are proven for certain classes of sets.