Collected papers on Ricci flow
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Seller Inventory n. Collected Papers on Ricci Flow Vol Publisher: International Press of Boston , This specific ISBN edition is currently not available. View all copies of this ISBN edition:. Synopsis The Ricci flow is currently a hot topic at the forefront of mathematics research. Buy New Learn more about this copy. Customers who bought this item also bought. Stock Image. Collected Papers on Ricci Flow Vol 37 various contributors].
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Published by International Press of Boston New Hardcover Quantity Available: 1. Seller Rating:. Collected Papers on Ricci Flow Hardback [various contributors]. Collected Papers on Ricci Flow H. Published by International Press of Boston Inc. New Hardcover Quantity Available: 3.
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Ricci curvature flow offers a powerful tool for computing it. Intrinsic curvature flows have been used in Riemannian geometry in the past 50 years with great success. These flows deform a given Riemannian metric according to its curvature. Among the most famous ones are the Ricci flow and the Yamabe flow. Both can be used to design Riemannian metrics with special curvature properties. The Ricci flow deforms the Riemannian metric according to its Ricci curvature.
In particular, it can be used to find a metric with constant Ricci curvature. There is a simple physical intuition behind it. Given a compact manifold with a Riemannian metric, the metric induces the curvature function. If the metric is changed, the curvature will be changed accordingly. The metric can be deformed in the following way: at each point, locally scale the metric, so that the scaling factor is proportional to the curvature at the point.
After the deformation, the curvature will be changed. Repeating this deformation process, both the metric and the curvature will evolve like heat diffusion.
Eventually, the curvature function will become constant everywhere. Another intrinsic curvature flow is called Yamabe flow. It has the same physical intuition with the Ricci flow, except for the fact that it is driven by the scalar curvature instead of Ricci curvature. For two manifolds, the Yamabe flow is essentially equivalent to the Ricci flow.
But for higherdimensional manifolds, Yamabe flow is much more flexible than the Ricci flow to reach constant-scalar-curvature metrics. Due to the ability of intrinsic curvature flows on metric designs, two curvature flows have been recently introduced into the engineering fields: a discrete Ricci flow on surfaces and a discrete Yamabe flow on surfaces. Through these works, the power of curvature flows has been extended from pure theoretical study to solving practical problems. One should note that in engineering fields , manifolds are usually approximated using discrete constructions, such as piecewise linear meshes; in order to employ curvature flow to solve practical problems, we need to extend the theories of curvature flows from the smooth setting to the corresponding discrete setting, and we need to pay attention to the convergence of the latter to the former.
Based on the discrete theories and formula, one is allowed to design computer algorithms that can simulate and compute the flow.
3D Surface Representation Using Ricci Flow (Computer Vision) Part 1
The theory of intrinsic curvature flows originated from differential geometry and was later introduced into the engineering fields. In this section, we give a brief overview of the literature that is directly related to the two flows mentioned above. For each flow, we would introduce some representative work on two aspects: theories in the smooth setting and the discrete setting.
The Ricci flow was introduced by Richard Hamilton in a seminal paper for Riemannian manifolds of any dimension. The Ricci flow has revolutionized the study of geometry of surfaces and three-manifolds and has inspired huge research activities in geometry. Hamilton used the twodimensional 2D Ricci flow to give a proof of the uniformization theorem for surfaces of positive genus.
This leads to a way for potential applications to computer graphics.
There are many ways to discretize smooth surfaces. The one that is particularly related to a discretization of conformality is the circle packing metric introduced by Thurston The notion of circle packing has appeared in the work of Koebe Thurston conjectured that for a discretization of the Jordan domain in the plane, the sequence of circle packings converges to the Riemann mapping. This was proved by Rodin and Sullivan This paved a way for a fast algorithmic implementation of finding the circle packing metrics, such as the one by Collins and Stephenson They proved a general existence and convergence theorem for the discrete Ricci flow and proved that the Ricci energy is convex.
An amendment to this paper has been published and can be accessed via a link at the top of the paper. Bhowmick, S. Clustering and summarizing protein-protein interaction networks: A survey. IEEE Trans. Data Eng. Yang, Z. A comparative analysis of community detection algorithms on artificial networks.
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Linköping University Conference Proceedings
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