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Conformal Geometry


Tuesday, 1 October 2019 to Sunday, 15 March 2020
Total hours: 40
Hours of lectures: 40

Examination procedure

  • Report or seminar
  • oral exam


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

Moser-Trudinger inequality

Uniformization Problem

Sobolev quotient on Rn

Yamabe problem

Time permitting: Nirenberg problem and prescription of scalar curvature


Educational goals:


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.

Bibliographical references

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.