我們將介紹幾何分析的重要問題與其應用。
<br>訓練學生思維能力。
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<br>The main goal of this course is to  introduce the existence of solutions to
<br>partial differential equations over manifolds.
<br>We will study the general theory of elliptic differential operators over Reimannian manifolds and 
<br>some connections with topology and differential geometry. 
<br>We will introduce Sobolev inequalities, techniques of  linear( or nonlinear )PDE and  some 
<br>applications to PDE .
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<br>1.Riemannian geometry :Metrics, Levi-Civita connection, curvature, Geodesics, Normal 
<br>coordinates, Riemannian Volume form.
<br>2.The Laplace equation on  manifolds :Existence, Uniqueness, Sobolev
<br>spaces, Schauder estimates,
<br>3. Hodge theory
<br>4.Elliptic equations and Uniformization theorem
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<br>References:
<br>Jurgen Jost, Riemannian Geometry and Geometric Analysis, Springer 2011.
<br>Peter Li, Geometric Analysis, Cambridge Univ. Press 2012. 
<br>Peter Petersen, Riemannian Geometry, Springer 2006.
<br>F. John: Partial Differential Equations. 4th Edition. 
<br>L. Evans: Partial Differential Equaitons
<br>Warner: Foundations of Differential Manifolds and Lie Groups. 
<br>Do Carmo: Riemannian Geometry. 
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<br>Exam 50%(to be scheduled)+ Homework 50%