课程名称: 体积泛函的变分和拉格朗日子流形
授课人： 陈竞一教授 （英属哥伦比亚大学，加拿大）
授课人简介： 陈竞一教授于1992年从斯坦福大学获得博士学位，随后任教于加州大学尔文分校、西北大学、麻省理工学院， 曾是斯坦福大学、布朗大学、普林斯顿大学访问教授。获得过美国和加拿大的多项研究奖项。
授课人主页： http://www.math.ubc.ca/~jychen
授课方式：线上线下同步授课。
授课时间：
2022年10月30日（周日） 8:00-10:45
2022年11月06日（周日） 8:00-10:45
2022年11月13日（周日） 8:00-10:45
2022年11月20日（周日） 8:00-10:45
2022年11月27日（周日） 8:00-10:45
2022年12月04日（周日） 8:00-10:45
授课地点：电子楼104
腾讯会议：605-7578-8530 会议密码：2022 （六次课相同）
课程简介：
Outline: We will begin with discussion on volume of submanifolds in a Riemannian manifold. Basic concepts such as the second fundamental form, the mean curvature vector of a submanifold will be introduced. We will study the critical points of the volume functional acting on submanifolds and their Euler-Lagrange equation. After this general setup, we will move to the special class of submanifolds, namely the so-called Lagrangian submanifolds in the complex Euclidean n-space and generalization to Kahler and symplectic manifolds. The Lagrangian submanifolds have their origins in Hamiltonian mechanics, and we will investigate the equilibriums of the volume within the class of Lagrangian submanifolds and the partial differential equations govern them.
Topcis:
Riemannian Geometry of Submanifolds
Geodesics and Minimal Submanifolds
Minimizing Volume via Calibration and Lagrangian Submanifolds
Brief Introduction to Symplectic Manifolds and Kahler Manifolds
Minimal Lagrangian Submanifolds
Hamiltonian Variations
Volume Critical Points under Hamiltonian Variations
Analysis of Partial Differential Equations for Critical Points of Volume of Lagrangian Submanifolds
References:
[1]Jingyi Chen, Lecture notes available at https://personal.math.ubc.ca/~jychen/MATH602-2021/Math602D-2021.pdf
[2]John Lee, Introduction to Smooth Manifolds (Graduate Texts in Mathematics, Vol. 218)
[3]M. Do Carmo, Riemannian Geometry, Birkhauser, Boston-BaselBerlin, 1992.
