The main theme of the current course is to study hypersurfaces in a Riemannian manifold, especially the level set of a potential function. A reference book is Peter Li's Geometric Analysis. We will introduce
area variation and mean curvature
second variation and this classical beautiful paper
volume comparison (6 hours maybe) and Cheeger-Gromoll splitting theorem
parabolic PDE and mean curvature flow
Some Reference: R. Schoen and S.-T. Yau, Lectures on Differential Geometry, Chapter 1.
This is an advanced course in differential geometry. If you are considering to take this course, please contact me as early as possible.
Here are some materials for you to prepare for this course:
The recording and notes of the introductory course Differential Geometry (85% English+15% Mandarin).
This course provides an introduction to the following three subjects:
Differential Geometry (manifolds, Lie derivatives, connections)
Riemannian Geometry (Levi-Civita connection, curvature, geodesics)
Geometric Analysis (Jacobi field, variations of length and energy)
Main Reference: Do Carmo, Riemannian Geometry.
Part of this course will be devoted to a review of Differential Geometry, conducted by our TA, Michael Tsai. However, it remains your responsibility to acquire all the required knowledge on your own.