Geometry II, Spring 2020

Course Information

Ceiba (link retired)
Lectures: Wednesday, 10:20 ~ 11:30, and Friday 10:20 ~ 11:30 at Astro-Math 101
Office hours: 14:00 ~ 15:00 every Tuesday, at Astro-Math 458
Grading scheme:
- Homework 40%
  You have two jokers: the lowest two grades will be discarded.
- Midterm 30%
- Final 30%
Graders: 李宸寬 and 郭茗崴
Course prerequisite:
- (undergraduate) analysis and algebra
- geometry of surfaces in the 3-space
- manifold, differential form
Main Topics:
- minimal surface
- vector bundle: connection, curvature and characteristic class
- basic theory for complex manifold
Textbooks:
- [L] H. Blaine, Jr. Lawson, Lectures on minimal submanifolds. Vol. I. mathscinet
- [S] Richard M. Schoen, Topic class on minimal surfaces. (notes taken by Xin Zhou)
- [M] Shigeyuki Morita, Geometry of differential forms. mathscinet
- [K] James Morrow and Kunihiko Kodaira, Complex manifolds. mathscinet

(Week 1) Review on the geometry of surfaces. First variational formula. note
(Week 2) Weierstrass representation. Simons identity. note
(Week 3) monotonicity formula, volume minimizing property of entire graph, second variational formula, stability. note
(Week 4) isoperimetric inequality. note
(Week 5) some phenomena in higher codimensions. note
(Week 6) differential forms, Stokes theorem, de Rham cohomology. note
(Week 7) vector bundles. note
(Week 8) connection, curvature. note
(Week 9) Pontryagin class and Chern class. note
(Week 10) Euler class. note
(Week 11) introduction to complex manifold. note
(Week 12) some examples of complex manifolds. note
(Week 13) Hermitian metric, Chern connection, Kähler metric. note
(Week 14) curvature of Kähler metric, Ricci curvature and canonical bundle. note
(Week 15) Hodge theory for Kähler manifolds. note
Kodaira Bochner formula, Kodaira vanishing theorem. note