A sharp quantitative stability result near infinitely concentrated minimisers
We consider the question of quantitative stability of minimisers for a well-known variational problem for which the infimum of the energy is not achieved in the classical sense, namely for the Dirichlet energy of degree $1$ maps from closed surfaces $(\Sigma,g_{\Sigma})$ of positive genus into the unit sphere $S^2\subset \mathbb{R}^3$. For this variational problem it is natural to view configurations which consist of a constant map from the given domain and an infinitely concentrated rotation as generalised minimisers and to hence ask whether the distance of almost minimisers $v:\Sigma\to S^2$ to this set of infinitely concentrated minimisers can be controlled in terms of the energy defect $\delta_v=E(v)-\inf E=E(v)-4\pi$. In this paper we develop a new dynamic approach that allows us to change the topology of the domain in a well controlled manner and to deform almost minimising maps from surfaces of general genus into harmonic maps from the sphere in a way that yields sharp quantitative estimates on all key features that characterise the distance to the set of infinitely concentrated minimisers, i.e. the scale of concentration, the $H^1$-distance to the nearest bubble on the concentration region and the $H^1$-distance to the nearest constant away from the concentration point.