arXiv research

For an embedded submanifold $Σ\subset\mathbb{R}^{N}$, Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds $Σ$ in the same way. For this purpose…

New method calculates Ricci curvature from distances between weighted volumes.

Study on modified Ricci curvature on graphs, proving rigidity and deriving formulas.

The problem of defining correctly geometric objects such as the curvature is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor…

The geometry of a ball within a Riemannian manifold is coarsely controlled if it has a lower bound on its Ricci curvature and a positive lower bound on its volume. We prove that such coarse local geometric control must persist for a definite amount of time under three-dimensional Ricci flow, and leads to local C/t deca…

We use the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. This function can be used to recover the Ricci tensor on smooth Riemannian manifolds by the formu…

Novel coarse extrinsic curvature for Riemannian submanifolds.

We introduce a novel definition of curvature for hypergraphs, a natural generalization of graphs, by introducing a multi-marginal optimal transport problem for a naturally defined random walk on the hypergraph. This curvature, termed \emph{coarse scalar curvature}, generalizes a recent definition of Ricci curvature for…

We define the distance between edges of graphs and study the coarse Ricci curvature on edges. We consider the Laplacian on edges based on the Jost-Horak's definition of the Laplacian on simplicial complexes. As one of our main results, we obtain an estimate of the first non-zero eigenvalue of the Laplacian by the Ricci…

Geometric sampling of networks using curvature measures.

We have performed an empirical comparison of two distinct notions of discrete Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci curvature were developed based on different properties of the classical smooth notion…

Ricci curvature was proposed by Ollivier in a general framework of metric measure spaces, and it has been studied extensively in the context of graphs in recent years. In this paper we prove upper bounds for Ollivier's Ricci curvature for bipartite graphs and for the graphs with girth at least 5. We also prove a genera…

A coupling method and an analytic one allow us to prove new lower bounds for the spectral gap of reversible diffusions on compact manifolds. Those bounds are based on the a notion of curvature of the diffusion, like the coarse Ricci curvature or the Bakry--Emery curvature-dimension inequalities. We show that when this …

HLRC offers a new curvature metric for hypergraphs that balances interpretability and efficiency.

We are concerned about the coarse and precise aspects of a priori estimates for Green's function of a regular domain for the Laplacian-Betrami operator on any $3\le n$-dimensional complete non-compact boundary-free Riemannian manifold through the square Sobolev/Nash/logarithmic-Sobolev inequalities plus the rough and s…

Abstract machinery finds obstructions to uniform positive scalar curvature.

Block and Weinberger show that an arithmetic manifold can be endowed with a positive scalar curvature metric if and only if its $\rationals$-rank exceeds 2. We show in this article that these metrics are never in the same coarse class as the natural metric inherited from the base Lie group. Furthering the coarse $C^\as…

Study compares metrics from negative curvature and quasi-Fuchsian representations.

We consider the class of compact n-dimensional Riemannian manifolds with cylindrical boundary, Ricci curvature bounded below by a given constant and injectivity radius bounded below by a positive constant, away from the boundary. For a manifold M of this class, we introduce a notion of discretization, leading to a grap…

New scalar curvature defined from Ollivier-Ricci curvature for graphs.

Combination theorem for geodesic coarsely convex group pairs.

Develops a smooth operator framework for analyzing neural network representations.

Survey paper analyzes curvature estimates for 4D gradient Ricci solitons.

Defines projective Ricci curvature and proves rigidity for sprays.

Paper classifies Randers metrics based on Ricci curvature properties.

Study shows central limit theorem for counting measures in non-smooth spaces.

Equivalence proven between different Ricci curvature definitions.

Defined Ricci curvature on simplicial complexes and proved bounds.

We introduce some new curvature quantities such as conformal Ricci curvature and bi-Ricci curvature and extend the classical Myers theorem under these new curvature conditions. Moreover, we are able to obtain the Myers type theorem for minimal submanifolds in ambient manifolds with positive bi-Ricci curvature. Some top…

Proves metrics with positive intermediate Ricci curvature on complex manifolds.

New discrete Ricci curvature for directed networks developed.

Study classifies Ricci solitons with specific curvature conditions.

The paper provides estimates for higher-order Ricci curvature along Kähler-Ricci flows.

Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.

The paper extends Ricci curvature in Finsler geometry and finds conditions for specific metrics.

Combinatorial approach to $α$-Ricci and Lin-Lu-Yau Ricci curvatures on graphs

For all complex dimensions n>=2, we construct complete Kaehler manifolds of bounded curvature and non-negative Ricci curvature whose Kaehler--Ricci evolutions immediately acquire Ricci curvature of mixed sign.

New curvature defined via graph resistances leads to Ricci flow.

The paper classifies certain types of Ricci solitons with specific curvature properties.

Three-manifolds with non-negative pinched Ricci curvature have complete Ricci flows.

Introduces new curvature concept for Kähler manifolds.

We show that in dimension 4 and above, the lifespan of Ricci flows depends on the relative smallness of the Ricci curvature compared to the Riemann curvature on the initial manifold. We can generalize this lifespan estimate to the local Ricci flow, using which we prove the short-time existence of Ricci flow solutions o…

The paper explores curvatures on graphs and their implications for Ricci flatness.

Ricci flow controls curvature on manifolds with bounds.

Positive Ricci curvature achieved by adding edges in a graph.

The paper studies weighted Ricci curvatures and characterizes Randers metrics.

Study on curvature decay in steady Ricci solitons, proving dichotomy.

Study on a weaker curvature condition for Kähler manifolds.