Pure and Applied Mathematics Quarterly

Volume 9 (2013)

Number 3

Visualization of 2-dimensional Ricci flow

Pages: 417 – 435

DOI: http://dx.doi.org/10.4310/PAMQ.2013.v9.n3.a2

Authors

Junfei Dai (Zhejiang University)

Xianfeng Gu (Stony Brook University)

Wei Luo (Zhejiang University)

Shing-Tung Yau (Harvard University)

Min Zhang (Stony Brook University)

Abstract

The Ricci flow is a powerful curvature flow method, which has been successfully applied in proving the Poincaré conjecture. This work introduces a series of algorithms for visualizing Ricci flows on general surfaces. The Ricci flow refers to deforming a Riemannian metric on a surface proportional to its Gaussian curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature on the surface converges to a constant function. Furthermore, during the deformation, the conformal structure of the surfaces is preserved. By using Ricci flow, any closed compact surface can be deformed into either a round sphere, a flat torus or a hyperbolic surface.

This work aims at visualizing the abstract Ricci curvature flow partially using the recent work of Izmestiev [8]. The major challenges are to represent the intrinsic Riemannian metric of a surface by extrinsic embedding in the three dimensional Euclidean space and demonstrate the deformation process which preserves the conformal structure. A series of rigorous and practical algorithms are introduced to tackle the problem. First, the Ricci flow is discretized to be carried out on meshes. The preservation of the conformal structure is demonstrated using texture mapping. The curvature redistribution and flow pattern is depicted as vector fields and flow fields on the surface. The embedding of convex surfaces with prescribed Riemannian metrics is illustrated by embedding meshes with given edge lengths. Using the variational principle developed by Izemestiev, the embedding is realized by optimizing a special energy function, the unique optimum is the solution.

Ricci flow can serve as a novel tool for visualization. It has broad applications, such as surface parameterization, shape design by curvature, shape deformation and shape analysis.

Keywords

Poincaré conjecture, uniformization metric, curvature, metric, surface parameterization

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