Non-univalent harmonic maps homotopic to diffeomorphisms

Abstract

We solve in this paper Problem 111 of the list compiled by S.-T.Yau in [32].Here is a restatement of this problem.Problem 111 of [32].Let / : Mi -> M 2 be a diffeomorphism between two compact manifolds with negative curvature.If h : M\ -> Mi is a harmonic map which is homotopic to /, is h a univalent map? (This problem has recently been reposed in [31] as Grand Challenge Problem 3.6.)The answer to the problem was proven to be yes when dimMi = 2 by Schoen-Yau [29] and Sampson [27].Part of the interest in the problem comes from the fact that harmonic maps have become extremely useful in proving rigidity results; see for example [30], [6],[14], [33], [15] and [20].Hence the negative answer given in this paper to Problem 111 places some limits on the applicability of the harmonic map techniques to rigidity questions.Our precise result is that for every integer n > 6 there is a pair of closed negatively curved Riemannian manifolds Mi and M 2 with dimMi = n, a diffeomorphism / : M\ -> M2, and a harmonic map h : Mi -> M 2 homotopic to / such that h is not univalent (i.e., not a one-to-one map).Furthermore given any e > 0, Mi and M2 can be constructed so that the sectional curvatures of M 2 are all pinched within e of -1 and Mi has constant -1 sectional curvatures.This paper has evolved from the earlier papers [9], [21], [10], [11] and [12].In fact, the crucial use made here of the Scharlemann-Siebenmann C°°-Hauptvermutung [28] was earlier used in [12].The second key ingredient is the existence of closed (real) hyperbolic manifolds with interesting cup product properties.Such manifolds are constructed in section

Locations

• Journal of Differential Geometry - View - PDF