Examples of Non‐Homeomorphic Harmonic Maps Between Negatively Curved Manifolds

Let M and N be closed non‐positively curved manifolds, and let f : M → N be a smooth map. Results of Eells and Sampson [ 1 ] show that f is homotopic to a harmonic map φ, and Hartman [ 6 ] showed that this φ is unique when N is negatively curved and f ∗(π 1 M ) is not cyclic. Lawson and Yau conjectured that if f : M → N is a homotopy equivalence, where M and N are negatively curved, then the unique harmonic map φ homotopic to f would be a diffeomorphism. Counterexamples to this conjecture appeared in [ 2 ], and later in [ 7 ] and [ 5 ]. There remains the question of whether a ‘topological’ Lawson–Yau conjecture holds. 1991 Mathematics Subject Classification 53C20, 55P10, 57C25, 58E20.