顧顯峰教授、羅鋒教授、丘成桐院士\ldots 等，在 2007年時提出了一種里奇流的離散方法並用這種方法找到相對應的保角映射。我們將在這篇論文中，說明並實作這種方法將虧格為 1 的封閉曲面保角地嵌入二維平面、里奇流何從連續到離散、和保角映射的關係。在算法的部份，分兩方面進行。第一部分，利用牛頓法解離散里奇流組成的非線性系統。並由之前理論的基礎知道里奇流會有解，所以更進一步寫成時間獨立的方程組藉由牛頓法找解。牛頓法在收斂的表現相當傑出。第二部份，實作丘院士及顧教授書中的一種計算上更有效率的方法，找到嵌入$\mathbb{R}^2$後的座標位置。而整套里奇流過程中，疊代的方式找保角的座標雖然是很直覺的，卻沒有效率。所以這方法找座標，是大幅度的縮減整體的時間花費。接著，我們用了兩個方法確認保角性。最後，我們提出使用里奇流這套方法找保角映射的一些經驗與建議。

Gu, Luo, and Yau etc. provided a process to discretize the Ricci flow and find the conformal mapping in 2007. In this thesis, we introduce the method and implement its algorithms to embed a closed genus-1 surface (in 3-D) into the Euclidean plane conformally. The algorithm is composed of two parts. First, we find the solution of the Ricci flow with Newton's method. By the convergence theory, it is known that there is a solution of the Ricci equation. We can write the equation as a time-independent equation system which solved by Newton's method. Newton's method is excellent on the rate of convergence. Second, we implement an efficiency way by Yau and Gu to find the position of vertices on $\mathbb{R}^2$ since the iterative method is intuitive but inefficient. Hence, the efficient method brings more benefits. Next, we use two methods to estimate the conformal errors. Finally, we provide some advices about the whole procedure to find the conformal mapping.

1 Motivation and Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Preliminary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Gauss Curvature and Gauss-Bonnet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Ricci Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Ricci Flow Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Discrete Gaussian Curvature and Ricci Flow . . . . . . . . . . . . . . . . . . . . 8
3.2.1 Exponential Convergence Rate . . . . . . . . . . . . . . . . . . . 9
3.2.2 Ricci Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2.3 Solve the Discrete Ricci Flow with Newton’s Method . . . . . . . 13
3.2.4 Newton’s Method to Solve K(u) = 0 . . . . . . . . . . . . . . . . 17
3.3 Embedded in the Euclidean Plane . . . . . . . . . . . . . . . . . . . . . . 19
3.3.1 Planar Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3.2 Least Vertices Energy . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.3 Comparison of Two Methods . . . . . . . . . . . . . . . . . . . . 26
4 Conformal Error Analysis . . . . . . . . . . . . . . . . . . . . . . . 27
4.1 Conformal Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 The Discrete Method of the Conformal Factor . . . . . . . . . . . . . . . 28
4.3 Conformal map and Holomorphism . . . . . . . . . . . . . . . . . . . . . 28
4.4 The Discrete Method of the Holomorphic Map . . . . . . . . . . . . . . . 29
5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . 32
5.1 Discrete Ricci Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2 Embedded in an Euclidean Plane . . . . . . . . . . . . . . . . . . . . . . 34
5.3 Conformal Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 37
A Basic Geometric Definitions and Theorems . . . . . . . . . . . . . . . . . . . . . . . 40
B Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . . . 42
C Models: Meshes in R 3 , Embedded in R 2 , and Conformal Error Results . . . . . 43

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