Let M and N be closed non-positively curved manifolds,
and
let f[ratio ]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∗(π1M) is not cyclic.
Lawson and Yau conjectured that if f[ratio ]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.