The 25th International Congress of Mathematicians held in Madrid in 2006 confirmed that the Russian mathematician Grisha Perelman, who was awarded a Fields Medal, had solved the Poincare Conjecture (P...
Some of the more differential aspects of the nascent field of computational topology are introduced and treated in considerable depth. Relevant categories based upon stratified geometric objects are p...
Publisher Summary This chapter provides an overview of computational topology. The first usage of the term “computational topology” appears to have occurred in the dissertation of M. Mantyla. The foc...
Osoinach introduced a way to construct a 3-manifold which can be obtained by the same integral Dehn surgery on an infinite number of knots in the 3-sphere. Using it, he gave such a hyperbolic 3-manifo...
Let (M, g, edv) be a weighted Riemannian manifold evolving by geometric flow ∂g ∂t = 2h(t), ∂φ ∂t = ∆φ. In this paper, we obtain a series of space-time gradient estimates for positive solutions of a p...
In this work we show that by restricting the coordinate transformations to the group of time-independent coordinate transformations it is possible to derive the Ricci flow from the contracted Bianchi ...
In this paper, we study Ricci flow on compact manifolds with a continuous initial metric. It was known from Simon that the Ricci flow exists for a short time. We prove that the scalar curvature lower ...
In this paper, we prove local results for solutions to the Ricci flow (heat flow) whose speed (height) is bounded by c/t for some time interval t ∈ (0, T). These results are contained in Chapter 7 of ...
In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let $g(t)$ be a smooth complete solution to the Ricci flow on...
In this note, we prove triviality and nonexistence results for gradient Ricci soliton warped metrics. The proofs stem from the construction of gradient Ricci solitons that are realized as warped produ...
In the present paper, we study basic properties of digital n-dimensional manifolds and digital simply connected spaces. An important property of a digital n-manifold is that M is a digital n-sphere if...
In machine learning, a high dimensional data set such as the digital image of a human face is often viewed as a point set distributed on a differentiable manifold. In many cases, the intrinsic dimensi...
In earlier work, carrying out numerical simulations of the Ricci flow of families of rotationally symmetric geometries on $S3$, we have found strong support for the contention that (at least in the ro...
In Ricci flow theory, the topology of Ricci soliton is important. We call a metric quasi-Einstein if the m-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalizatio...
I discuss certain applications of the Ricci flow in physics. I first review how it arises in the renormalization group (RG) flow of a nonlinear sigma model. I then review the concept of a Ricci solito...
For a closed smooth manifold M , we consider a closure of the set of metrics on M with sectional curvature bounded between −1 and 1. We introduce a variant of Gromov’s minimal volume, called essential...
Background from Riemannian geometry and Ricci flow: Preliminaries from Riemannian geometry Manifolds of non-negative curvature Basics of Ricci flow The maximum principle Convergence results for Ricci ...
Algebra.- Topology.- Manifolds.- Bundles and Connections.- Characteristic Classes.- Theory of Fields, I: Classical.- Theory of Fields, II: Quantum and Topological.- Yang-Mills-Higgs Fields.- 4-Manifol...
Abstract Let ( M , g ) be an n-dimensional Riemannian manifold with a quasi-Einstein metric g, i.e., g satisfies the following equation Ric + Hess f − 1 τ d f ⊗ d f = λ g for constants τ > 0 and λ, he...