In this survey we review some results concerning negatively curved
exotic structures (DIFF and PL) and its (unexpected)
implications on the limitations of some analytic methods in geometry.
Among these methods are the harmonic map method and the Ricci flow
method.
First in section 1 we mention certain results about the rigidity of
negatively curved manifolds. In section 2 and 3 we survey some results
concerning the limitations of the harmonic map technique and the
natural map technique for negatively curved manifolds. Finally,in
section 4, we mention some limitations of the Ricci flow method
for pinched negatively curved manifolds.
We are grateful to J-F. Lafont and R. Spatzier for the useful
information they provided to us and to E.Gasparim for suggesting some
improvements in the text. We are also grateful to the referee for
pointing out certain inaccuracies.