  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.

Analysis of PDEs;Differential Geometry