Exotic Structures and the Limitations of Certain Analytical Methods in Geometry

Abstract

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.