Related papers: Recent rigidity results for graphs with prescribed…
A rigidity theory is developed for bar-joint frameworks in linear matrix spaces endowed with a unitarily invariant norm. Analogues of Maxwell's counting criteria are obtained and minimally rigid matrix frameworks are shown to belong to the…
In this article, we give results of prescribing scalar and mean curvature functions for metrics either pointwise conformal or conformally equivalent to a Riemannian metric that is equipped on a compact manifold with boundary, with…
In a previous work, the authors have introduced a Lin-Lu-Yau type Ricci curvature for directed graphs, and obtained a diameter comparison of Bonnet-Myers type. In this paper, we investigate rigidity properties for the equality case, and…
If a graph submanifold $(x,f(x))$ of a Riemannian warped product space $(M^m\times_{e^{\psi}}N^n,\tilde{g}=g+e^{2\psi}h)$ is immersed with parallel mean curvature $H$, then we obtain a Heinz type estimation of the mean curvature. Namely, on…
We study the set of curvature functions which a given compact manifold with boundary can possess. First, we prove that the sign demanded by the Gauss-Bonnet Theorem is a necessary and sufficient condition for a given function to be the…
Discrete curvatures are quantities associated to the nodes and edges of a graph that reflect the local geometry around them. These curvatures have a rich mathematical theory and they have recently found success as a tool to analyze networks…
We prove that on a closed surface, for any $c>0$, our min-max theory for prescribing mean curvature produces a solution given by a curve of constant geodesic curvature $c$ which is almost embedded, except for finitely many points, at which…
In this paper, we prove a rigidity theorem for Poincar\'e-Einstein manifolds whose conformal infinity is a flat Euclidean space. The proof relies on analyzing the propagation of curvature tensors over the level sets of an adapted boundary…
The definition of Ricci curvature on graphs in Bakry-\'Emery's sense based on curvature dimension condition was introduced by Lin and Yau [\emph{Math. Res. Lett.}, 2010]. Hua and Lin [\emph{Comm. Anal. Geom.}, 2019] classified unweighted…
Hyperfiniteness or amenability of measurable equivalence relations and group actions has been studied for almost fifty years. Recently, unexpected applications of hyperfiniteness were found in computer science in the context of testability…
In this paper, we develop a min-max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth,…
Lott-Sturm-Villani theory of curvature on geodesic spaces has been extended to discrete graph spaces by C. L{\'e}onard by replacing W2-Wasserstein geodesics by Schr{\"o}odinger bridges in the definition of entropic curvature [23, 25, 24].…
In this paper, we show that the Ricci curvature lower bound in Ollivier's Wasserstein metric sense of a continuous time jumping Markov process on a graph can be characterized by some optimal coupling generator and provide the construction…
It is proved the existence and uniqueness of graphs with prescribed mean curvature in Riemannian submersions fibered by flow lines of a vertical Killing vector field.
A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint…
We introduce a curvature function for planar graphs to study the connection between the curvature and the geometric and spectral properties of the graph. We show that non-positive curvature implies that the graph is infinite and locally…
Let $M$ be a compact connected surface with boundary. We prove that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function $f$ on $\partial M$ (resp. on $M$) to be geodesic curvature…
We consider the graphical mean curvature flow of maps ${\bf f}:\mathbb{R}^m\to\mathbb{R}^n$, $m\ge 2$, and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed…
In this paper, we investigate the rigidity problems of complete hypersurfaces with constant mean curvature and constant scalar curvature in Euclidean spaces. Firstly, under some conditions of Gaussian-Kronecker curvature, we provide…
In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence…