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研究生： 李佳穎
Lee, Chia-Ying
論文名稱： 加權流形上的幾何
Geometry of Weighted Manifolds
指導教授： 宋瓊珠
Sung, Chiung-Jue Anna
口試委員: 邱鴻麟
Chiu, Hung-Lin
蕭育如
Syau, Yu-Ru
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2020
畢業學年度： 108
語文別： 英文
論文頁數： 225
中文關鍵詞： 光滑賦距測度空間 、巴克里-埃默瑞曲率 、加權流形 、熱方程 、拉普拉斯方程 、梯度估計 、熱核 、格林函數 、最低頻譜 、分裂定理 、帕松方程 、龐加萊不等式 、黎曼流形 、瑞奇孤立子 、哈奈克不等式
外文關鍵詞： smooth metric measure space, Bakry-Émery curvature, weighted manifold, heat equation, Laplacian equation, gradient estimate, heat kernel, Green's function, bottom spectrum, splitting theorem, Poisson equation, Poincaré inequality, Riemannian manifold, Ricci soliton, Harnack inequality
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    • 在這篇論文中，我們關注在伴隨某種增長加權函數的光滑賦距測度空間上的幾何。首先在滿足加權龐加萊不等式的完備黎曼流形上，我們學習了調和函數的衰退估計和體積估計，並藉此了解在這種流形上的分裂定理。接著我們聚焦於伴隨有界梯度的加權函數和有負下界巴克里-埃默瑞曲率的加權流形並考慮其上的加權熱方程。我們建立了此熱方程正解的梯度估計並給出熱核和格林函數的估計。最後我們學習將其應用於在伴隨正頻譜加權流形上的帕松方程及穩定瑞奇孤立子。

    In this thesis, we focus on the geometry of the smooth metric measure space (M,g,exp(-f)dv) with some growth weighted function f. We first study the decay estimates of harmonic functions and volume estimates on complete Riemannian manifolds satisfying a weighted Poincaré inequality. Then we study splitting theorems on such manifolds. Next, we consider the weighted heat equation on manifolds with bounded gradient f and Bakry-Émery curvature bounded from below. In particular, we establish a gradient estimate of the positive solutions and give estimates of the heat kernel and the Green's function. Finally, we study the weighted Poisson equation and steady Ricci solitons on such manifolds with positive spectrum as applications.

    Abstract...........................................................i Acknowledgement...................................................ii Contents.........................................................iii 1 Introduction.....................................................1 2 Preliminaries...................................................11 2.1 Definition....................................................11 2.2 First eigenvalue..............................................15 2.3 Bochner formula...............................................16 2.4 Laplacian comparison theorem..................................18 2.5 Volume comparison theorem.....................................23 3 Complete Manifolds with Positive Spectrum.......................26 3.1 Preliminary estimates.........................................26 3.2 Splitting theorem.............................................39 4 Complete Manifolds with Weighted Poincaré Inequality............58 4.1 Existence of the weight function ρ(x).........................59 4.2 Preliminary estimates.........................................64 4.3 A new type Bochner formula....................................78 4.4 Splitting theorem.............................................86 5 Some Structure Theorems on Smooth Metric Measure Spaces........101 5.1 Volume upper and lower bounds................................101 5.2 Splitting for linear weight..................................123 5.3 Splitting for quadratic weight...............................132 6 Weighted Heat Equation on Manifolds with Bounded Gradient f....139 6.1 Gradient estimate for f-Laplacian equation...................140 6.2 Gradient estimate for f-heat equation........................148 6.3 Heat kernel estimate.........................................156 6.4 Green’s function estimate....................................165 7 Existence and Estimate of Weighted Poisson Equation on Manifolds .................................................................170 7.1 Preliminary estimates........................................170 7.2 Solving weighted Poisson equation............................197 8 Applications to Steady Ricci Solitons..........................205 8.1 Heat kernels on steady Ricci solitons........................206 8.2 Poisson equations on steady Ricci solitons...................207 Appendix.........................................................215 A. Gradient estimate on complete manifolds.......................215 B. Sobolev inequality............................................219 C. Stoke’s Theorem...............................................219 D. Semi-group property for heat kernels..........................221 Bibliography.....................................................224

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