Book Recommendation - PDE's for geometricians / topologists 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.
 A: Lectures on Partial Differential Equations by V. Arnold. 
A: Since you claim that you know "nothing about PDEs", I think it would be very hard to appreciate the topological/geometric applications of PDEs without at least a basic familiarity with the theory of PDEs: i.e. methods used for proving existence, uniqueness, regularity and basic estimates in time/space. I would say start with the PDE lectures by V. Arnold, and read it along with Evans' PDE text. 
A: 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.
A: Micheal Taylor's series has already been mentioned. You might also look at Thierry Aubin, Some Nonlinear Problems in Riemannian Geometry, which is a good guide for the use of nonlinear elliptic PDE in geometry. I took a reading course from this book when I was a Ph.D. student.
A: Riemannian Geometry and Geometric Analysis by Jürgen Jost might be helpful.
