N ov 2 00 3 Harmonic Cellular Maps which are not Diffeomorphisms

The use of harmonic maps has been spectacularly successful in proving rigidity (and superrigidity) results for non-positively curved Riemannian manifolds. This is witnessed for example by results of Sui [31], Sampson [26], Corlette [6], Gromov and Schoen [18], Jost and Yau [22], and Mok, Sui and Yeung [23]. All of which are based on the pioneering existence theorem of Eells and Sampson [13] and the uniqueness theorem of Hartman [19] and Al’ber [1]. In light of this we believe that it is interesting to demarcate this technique. For example, it was shown in [15] that a harmonic homotopy equivalence between closed negatively curved Riemannian manifold is sometimes not a diffeomorphism, even when one of the manifolds has constant sectional curvature equal to −1 (i.e. is a real hyperbolic manifold). Later other examples were given in [16] and [17] where such a harmonic homotopy equivalence f is not even a homeomorphism; even though the ones constructed in [17] are homotopic to diffeomorphisms. In this paper we construct a harmonic map h between closed negatively curved Riemannian manifolds M and N which is not a diffeomorphism but is the limit of a 1-parameter family of diffeomorphisms; in particular, h is a cellular map. In our example either M or N (but of course not both) can be a real hyperbolic manifold and the other have its sectional curvature pinched within ǫ of −1, where ǫ is any preassigned positive number. (We do not know whether such a harmonic map h can ever be a homeomorphism. See our acknowledgment below.) This result is contained in Theorem 1, its Addendum and Theorem 2. We construct such examples in all dimensions > 10 and conjecture that this can be improved to all dimensions ≥ 6. We have also discovered a curious relationship between the Poincaré Conjecture in low dimensional topology and the existence of a certain type of harmonic map k : M → N between ∗The first author was partially supported by a NSF grant. The second author was supported in part by a research grant from CNPq, Brazil.