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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2603.25361 |
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| _version_ | 1866912983717773312 |
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| author | Rupflin, Melanie Woodward, Sebastian |
| author_facet | Rupflin, Melanie Woodward, Sebastian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25361 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A sharp quantitative stability result near infinitely concentrated minimisers Rupflin, Melanie Woodward, Sebastian Analysis of PDEs Differential Geometry 58E20, 53C43, 26D10, 49Q05 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. |
| title | A sharp quantitative stability result near infinitely concentrated minimisers |
| topic | Analysis of PDEs Differential Geometry 58E20, 53C43, 26D10, 49Q05 |
| url | https://arxiv.org/abs/2603.25361 |