TITLE: Book Recommendation - PDE's for geometricians / topologists
QUESTION [24 upvotes]: I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as more "qualitative" methods, rather then approximations or techniques for solutions.
I am looking for anything that might help me to study this, books, papers, surveys, etc.
What is your recommended road map for familiarizing myself with this subject?
Thank you.

REPLY [8 votes]: I find your question too broad. I would recommend starting with a book that focuses on a particular question or area in differential geometry and presents the PDE theory needed. A very incomplete list of suggestions include the following:

Aubin's book is a good way to learn the PDE theory required for the Yamabe problem.
A long list of books on Yang-Mills gauge theory can be found here:
http://www.amazon.com/Mathematical-Gauge-Theory-Books/lm/R2JAMJ9TVJ3DYU
Another list of books on Ricci flow here:
http://www.amazon.com/s/ref=nb_sb_noss_2?url=search-alias%3Daps&field-keywords=ricci%20flow

There are other books on the Atiyah-Singer index theorem, harmonic maps, minimal surfaces, the complex Monge-Ampere equation, etc.
Books on elliptic PDE's used by many differential geometers include Gilbarg-Trudinger and Morrey.