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A sharp quantitative stability result near infinitely concentrated minimisers

Melanie Rupflin, Sebastian Woodward

Abstract

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 $(Σ,g_Σ)$ 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 $v:Σ\to S^2$ to this set of infinitely concentrated minimisers can be controlled in terms of the energy defect $δ_v=E(v)-\inf E=E(v)-4π$. 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.

A sharp quantitative stability result near infinitely concentrated minimisers

Abstract

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 maps from closed surfaces of positive genus into the unit sphere . 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 to this set of infinitely concentrated minimisers can be controlled in terms of the energy defect . 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 -distance to the nearest bubble on the concentration region and the -distance to the nearest constant away from the concentration point.

Paper Structure

This paper contains 11 sections, 11 theorems, 175 equations.

Table of Contents

  1. Introduction and main results
  2. Definition and basic properties of metrics
  3. Łojasiewicz Estimates for maps with small energy defect
  4. Analysis of our new gradient flow
  5. Proof of Theorem \ref{['thm:main']}
  6. Proof of Theorem \ref{['thm:Loj-NHD']}
  7. Proof of the technical lemmas used in Section \ref{['sec:proof-LojEst']}
  8. Precise definition and key properties of the maps $\bar{z}_{\lambda,a}$
  9. Variations of $E(\bar{z}_{\lambda,a})$ with respect to $a$
  10. Proof of Lemma \ref{['lemma:special-variation']}
  11. Bounds on the second variation

Key Result

Theorem 1.1

Let $(\Sigma,g_\Sigma)$ be any closed oriented surface with positive genus. Then there exists a constant $C> 0$ such that the following holds true. For any degree $1$ map $v\in H^1(\Sigma, S^2)$, there exists a rotation ${\mathcal{R}} \in SO(3)$, a point $a\in \Sigma$, a value $p \in S^2$ and a scal

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • Corollary 2.5
  • proof : Proof of Proposition \ref{['prop:metrics-main']}
  • proof : Proof of Corollary \ref{['cor:proj-cylinder']}
  • Remark 3.1
  • ...and 22 more