Harmonic Maps with Fixed Singular Sets

§ 0. Introduction Here, for a smooth domain Ω in R and a compact smooth Riemannian manifold N we study a space H consisting of all harmonic maps u : Ω → N that have a singular set being a fixed compact subset Z of Ω having finite m− 3 dimensional Minkowski content. This holds if, for example, Z is m− 3 rectifiable [F, 3.2.14]. We define a suitable topology on H using Hölder norms on derivatives weighted by powers of the distance to Z. We study the structure of H and other properties, such as stability and minimality, under perturbations. For example, near a homogeneous harmonic map that is singular only at the origin, H is, for suitable powers of the weights, a smooth manifold [Theorem 4.7]. An application [Theorem 5.6] of the result to some well-known harmonic maps, such as the homogeneous extension of the harmonic maps from S to S, the identity from Sm−1 to itself and the equator maps from Sm−1 to S, gives that their boundary data can be perturbed in any directions of the eigenvectors corresponding to the positive eigenvalues (of the boundary data) to obtain new harmonic maps. These new harmonic maps are also stable (energy minimizing, unique) if the perturbed maps are strictly stable (strictly minimizing), which is implied by Theorems 3.5, 3.8 saying that stability and minimality are preserved under small perturbations. For smooth harmonic maps, we prove that H is a Banach manifold modelled on the space of boundary data, and the projection map that sends u to u|∂Ω is Fredholm of index 0 [Theorem 6.4]. Locally, near u ∈ H with K0 being the space of smooth Jacobi fields along u that vanish on ∂Ω, some neighborhood of u in H is diffeomorphic to a submanifold (of codimension dim (K0)) of the product of K0 with the space of the boundary data [Theorem 6.2]. Using the global structure theorem, we prove a generic uniqueness property of smooth harmonic maps [Theorem 6.8], which establishes the first category nature of the set of all boundary maps that occur as trace of two distinct harmonic maps having same energy. We prove a Schauder-type estimate and a pointwise estimate in Section 2 [Theorems 2.1, 2.3]. This is relevant for a study of the Fredholm property of the Jacobi operator, which is essential to apply the implicit function theorem. For isolated singularities, a very detailed analysis of this property with perturbation applications has recently been given by N. Smale [SN3]. If one allows the singularity to vary, there is the example of [HKL] where the boundary data corresponds under the sterographic projection to the conformal map z. Here there is a one parameter family of distinct harmonic maps having this Dirichlet boundary data and the same energy (by energy minimality). The singularity varies. This phenomenon holds for any n-axially symmetric boundary data of non-zero degree when n ≥ 2. (See [HKL].) For smooth case, we follow with some modifications some steps in Brian White’s study [WB] of the corresponding theory for smooth immersed minimal submanifolds. The presence of singularities demands various new estimates and leads to several interesting examples of singular Jacobi fields and families of harmonic maps. The authors would like to thank Frank Pacard and Nat Smale for their interest and for pointing out an error in an earlier version of this paper.