Measured Foliations and Harmonic Maps of Surfaces

Fix a Riemannian surface of negative curvature (N, h), and a differentiable surface M g of the same genus g that will host various structures. Also fix a diffeomorphism f0 : M → N . It is well known ([ES], [A], [SY1], [Sa]) that to every complex structure σ on M , there is a unique harmonic diffeomorphism f(σ) : M(σ) → (N, h) homotopic to f0 : M → N ; one is led to consider what other, possibly ostensibly weaker, structures on M might also determine harmonic maps from M to N homotopic to f0. The goal of this paper is to show (Theorem 3.1) that a harmonic map f(σ′) : (M, σ′) → (N, h) may be uniquely specified by the initial choice of a class of measured foliations (representing the maximal stretch measured foliation for the harmonic map f(σ′)) rather than an initial choice of complex structure: we observe that a measured foliation may be considered to be a differential-topological object in contrast to the analytical object that a complex structure σ represents. Our proof has aspects of independent interest. In particular, in the proof of uniqueness (§4), from a harmonic map f : (M, σ) → (N, h) of a surface, we construct a naturally associated equivariant (area) minimal map F : (M̃, σ̃) → (Ñ , h̃)× (T, 2d) of the universal cover into the product of the universal cover (Ñ , h̃) with a real tree (T, 2d). We show (Theorem 4.3) that for two dimensional negatively curved targets (N, h), that this minimal map is stable; we also develop some of the necessary background of this construction and result. There are a number of contexts for our result. We begin by recalling the Hodge-like theorem of Hubbard and Masur ([HM]; see [W5] for a Hodge-like proof) which states that on a given Riemann surface R, to each measured foliation (F , μ) there exists a