http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/66 2026-03-10T19:37:48Z http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/93761 標題: 黏性解與等高線流; Viscosity solutions and the level set flow 作者: 廖宣霖; Hsuan-Lin Liao 摘要: 給定以時間 t geq 0 為參數且在 R^{n+1} 中的超曲面 Gamma(t), n geq 2。如果 Gamma(t)上每點的垂直速率與該點的均曲率吻合，我們即說 Gamma(t) 依均曲率型變，或者說，Gamma(t)是均曲率方程的解。有趣的是，Gamma(t)可能在某個時間 t_0 產生奇異點，也就是說，Gamma(t) 在 t_0 時並非一光滑的超曲面。為了把 Gamma(t) 延拓過 t_0, 論文[2]中利用黏性解的水平集來定義均曲率方程的廣義解，我們稱其廣義解為廣義均曲率流或等高線流。此外，論文[1]中證明了一個有關旋轉對稱等高線流的一個漂亮定理，因此在本論中，我們將對於[2]中所定義的等高線流與[1]中的漂亮定理做一個介紹。; We say a family of smooth hypersurfaces Gamma(t) in R^{n+1}, n geq 2 evolve by mean curvature if its normal velocity is equal to its mean curvature, or say it is the solution to the mean curvature equation. However, Gamma(t) may develop singularities in finite time, that is, at some t_0<infty, Gamma(t_0) is not a smooth hypersurface. To extend the solution pass the singularity, Chen, Giga, Goto, [2] defined a generalized flow by level sets of viscosity solutions. We call this flow the generalized mean curvature flow or the level set flow. Moreover, in 1991, Altschuler, Angenent and Giga [1] proved a beautiful theorem regarding rotationally symmetric level set flows. Thus, in this thesis, we present the level set flow defined in [2], and the beautiful theorem in [1]. 2024-01-01T00:00:00Z http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/39440 標題: 黎曼流型上的熱核估計; Heat kernel estimate on Riemannian manifolds 作者: Jung-Sheng Lu; 呂融昇 摘要: 在一般的歐氏空間上，我們已經能清楚的把熱核表達出來。但在較為複雜的黎曼曲面上則難以表達。在我的文章中，我將會分為在完備而不緊緻的黎曼流型上以及完備且邊界是凸的黎曼流型上分別討論。在這兩種情況上，藉由得到相同的梯度估計，以此推導出相同的哈拿估計，在由此二估計，推出相同型式的熱核上界。另外，藉由畢社比較定理，估計出在此二種情況中相同型式的熱核下界。最後，利用熱的上界及下界可導出在拉普拉斯運算下的特徵值下界、以及格林函數的估計。; In the common Euclidean spaces, we can explicitly express the form of heat kernel.But in the more complicated Riemannian manifolds, it is hard to express. In my survey, I will discuss in two cases, first, in complete noncompact manifold, second, in compactmanifold with convex boundary. In this two cases, we can get the same form of gradient estimate and Harnack estimate, and by these two estimates, we can get the same form ofheat kernel upper bound in these two cases. Also, by Bishop volume comparison, we can get the same form of heat kernel upper bound in these two cases. With heat kernelestimates, we can estimate the lower bound of eigenvalue of Laplace operator and Green function. 2011-01-01T00:00:00Z http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/43376 標題: 黎曼曲面上之保角變換; Survey on conformal metrics with prescribed Gaussian curvature on compact Riemannian manifold 作者: Hung-Zen Liao; 廖宏仁 摘要: 有關黎曼曲面上，在給定高斯曲率後，是否存在保角變換，使得原來的黎曼度量跟後來的黎曼度量有這樣的保角關係，若否，是否能找的條件使其成立。; If g is a metric on M and if K' satisfies the sign conditions, is K' the curvature of some metric g', that is pointwise conformal to g. 2009-01-01T00:00:00Z http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/71225 標題: 黎曼—羅赫定理的一個代數方法之證明; A Proof of Riemann-Roch Theorem by Algebraic Methods 作者: Hao-Wei Huang; 黃皓偉 摘要: We start from some basic notions, like sheaves and cohomology, and try to introduce and prove Riemann-Roch theorem in the 2-dimension case. The definition of cohomology of a sheaf is more difficult to compute in some situation. However, the Čech cohomology of a sheaf over a paracompact space is isomorphic to the usual definition of cohomology,and Čech cohomology gives us a more concrete way to think what the cohomology of a sheaf is. In chapter 3 we introduce the concept of twisted complexes. We will use it to compute Ext and the class in Čech cohomology which is in the statement of Riemann-Roch theorem, and identify this class with characteristic class Td in cochain level by direct computation. 2018-01-01T00:00:00Z