# Low energy levels of harmonic maps into analytic manifolds

## TL;DR

This paper investigates the energy spectrum of harmonic maps into analytic manifolds, proving that for generic 3-manifolds, the spectrum's accumulation points are excluded, and establishing obstructions to harmonic sphere gluing.

## Contribution

It demonstrates that the lowest potential accumulation point of the energy spectrum is not realized for generic 3-manifolds and introduces new obstructions to harmonic sphere gluing.

## Key findings

- Excludes the possibility of energy spectrum accumulation at 2 E_{min} for generic 3-manifolds.
- Establishes obstructions to gluing harmonic spheres.
- Provides Lojasiewicz-estimates for almost harmonic maps.

## Abstract

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.