Good sources to learn about Geometric Analysis

So far the topics that have captured my interest the most are analysis and geometry. I want to learn a bit more about geometric and global analysis. There seems to be no shortage of geometry books but I was wondering what I should learn in the way of analysis. As of now I know some basic measure theory and functional analysis and was hoping for somebody to point me in the right direction. Thank you.

Solution 1:

Here are three (somewhat standard) references:

• Jurgen Jost, "Riemannian Geometry and Geometric Analysis."
• Peter Li, "Lecture Notes on Geometric Analysis."
• Thierry Aubin, "Nonlinear Analysis on Manifolds. Monge-Ampere Equations."

Jost's book is on its sixth edition. Aubin's book has a first and second edition, although my understanding is that the first edition might actually be more suitable.

There are also some specialized texts that are nevertheless still written in an accessible manner, and might serve as introductions to geometric analysis. Examples include:

• Ben Andrews, Christoffer Hopper, "The Ricci Flow in Riemannian Geometry."
• Peter Topping, "Lectures on Ricci Flow."
• Jerry Kazdan, "Applications of Partial Differential Equations to Problems in Geometry."

Most of the above books only assume standard courses in real analysis and Riemannian geometry as pre-requisites. In particular, I don't think that any PDE knowledge is explicitly assumed (except possibly for Kazdan's book).

Still, since you asked about analysis, here are some topics in analysis that I, as a beginner in geometric analysis, have found important:

• The Spectral Theorem (in functional analysis)
• Sobolev spaces
• The Laplace equation (PDE)
• The heat equation (PDE)
• Second-order elliptic equations (PDE)
• Calculus of variations
• Hausdorff measure and rectifiable sets (Geometric measure theory)

Again, you don't need to know any of these topics to begin reading, but they couldn't hurt.