Examples of Non-Homeomorphic Harmonic Maps Between Negatively Curved Manifolds

Let M and N be closed non-positively curved manifolds, and letf:MN be a smooth map. Results of Eells and Sampson 1] showthat f is homotopic to a harmonic map , and Hartman 6] showedthat this is unique when N is negatively curved and f_*(₁ M)is not cyclic. Lawson and Yau conjectured that if f:MN is ahomotopy equivalence, where M and N are negatively curved, thenthe unique harmonic map homotopic to f would be a diffeomorphism.Counterexamples to this conjecture appeared in 2], and laterin 7] and 5]. There remains the question of whether a ‘topological’Lawson–Yau conjecture holds. 1991 Mathematics SubjectClassification 53C20, 55P10, 57C25, 58E20.