Many interesting geometric objects are characterised as minimisers or critical points of a natural quantity and in this talk we will discuss harmonic maps from surfaces into Riemannian manifolds, i.e. critical points of the Dirichlet energy and consider two basic questions that arise for many variational problems:
On the one hand we ask whether knowing that the energy of a map is close to the minimal possible energy guarantees that the map is close to a minimiser.
On the other hand, if we consider more general critical points, then we may also ask whether knowing that the gradient of the energy is small guarantees that the map is close to a critical point and what this can tell us about the energy level of critical points.
In practice it is important to understand these questions not only at a qualitative, but also at a quantitative level, and we present new results on these questions for (almost) harmonic maps with small energy into analytic manifolds.