Low energy levels of harmonic maps into analytic manifolds

We consider the energy spectrum $\Xi_E(N)$ of harmonic maps from the sphere into a closed Riemannian manifold $N$. While a well known conjecture asserts that $\Xi_E(N)$ is discrete whenever $N$ is analytic, for most analytic targets it is only known that any potential accumulation point of the energy spectrum must be given by the sum of the energies of at least two harmonic spheres. The lowest energy level that could hence potentially be an accumulation point of $\Xi_E$ is thus $2 E_{min}$. In the present paper we exclude this possibility for generic 3 manifolds and prove additional results that establish obstructions to the gluing of harmonic spheres and Lojasiewicz-estimates for almost harmonic maps.