Prerequisites are Functional Analysis and Sobolev Spaces in Rn. It is useful some
basic knowledge of elliptic regularity theory and of Differential Geometry, but there
will be a recollection of some basic notions and results.
The course is intended for Ph.D. students and for undergraduates in their fifth year.
Brief review in Differential Geometry
Sobolev Spaces on Manifolds
Sobolev quotient on Rn
Time permitting: Nirenberg problem and prescription of scalar curvature
The aim of the course is to apply methods in functional analysis, calculus of variatons and
elliptic theory to geometric problems, in particular with lack of compactness.
T. Aubin: Some nonlinear problems in Riemannian geometry. SMN, Springer, 1998.
E. Hebey, Sobolev spaces on Riemannian manifolds. Lecture Notes in Mathematics. Springer, 1996.
J. Lee - T. Parker, The Yamabe problem. Bull. Amer. Math. Soc., 1987.