Gluing instantons à la Brezis–Coron in dimension four and the dipole construction

Luca Martinazzi; Tristan Rivière

doi:10.5802/crmath.795

Article de recherche - Équations aux dérivées partielles

Gluing instantons à la Brezis–Coron in dimension four and the dipole construction
[Recollement d’instantons à la Brezis–Coron en dimension quatre et la construction de dipôles]

Luca Martinazzi ¹ ; Tristan Rivière ²

¹ Department of Mathematics Guido Castelnuovo, Università di Roma La Sapienza, Italy
² Department of Mathematics, ETH Zentrum, 8093 Zürich, Switzerland

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1219-1261

Cet article fait partie du numéro thématique De génération en génération : l'héritage mathématique de Haïm Brezis coordonné par Henri Berestycki et al..

Abstract (VO)
Résumé

Given a connection $A$ on a $\operatorname{SU}(2)$-bundle $P$ over $\mathbb{R}^4$ with finite Yang–Mills energy $\operatorname{YM}(A)$ and nonzero curvature $F_A(0)$ at the origin, and given $\rho >0$ small enough, we construct a new connection $\hat{A}$ on a bundle $\hat{P}$ of different Chern class ($\left|c_2(A)-c_2(\hat{A}) \right|= 8\pi ^2$), in such a way that $\hat{A}$ is gauge equivalent to $A$ in $\mathbb{R}^4\setminus B_\rho (0)$, gauge equivalent to an instanton in a smaller ball $B_{\tau \rho }(0)$, and

\[ \operatorname{YM}(\hat{A})\le \operatorname{YM}(A)+8\pi ^2-\varepsilon _0\rho ^4\left|F_A(0) \right|^2, \]

where $\tau \in (0.3,0.4)$ and $\varepsilon _0>0$ are universal constants independent of $A$ and $\rho $. Our gluing method is similar in spirit to the one of Brezis–Coron for harmonic maps. We compare it with classical results by Taubes and discuss applications and open problems.

Étant donnée une connexion $A$ sur un fibré principal $P$ en groupe $\operatorname{SU}(2)$ sur $\mathbb{R}^4$ avec une énergie de Yang–Mills finie égale à $\operatorname{YM}(A)$ et ayant une courbure non nulle $F_A(0)$ à l’origine, et étant donné un rayon $\rho >0$ suffisamment petit, nous construisons une nouvelle connexion $\hat{A}$ sur un $\operatorname{SU}(2)$ fibré principal $\hat{P}$, de classe de Chern différente à celle de $P$ ($\left|c_2(A)-c_2(\hat{A}) \right|= 8\pi ^2$), de telle sorte que $\hat{A}$ soit équivalente de jauge à $A$ dans $\mathbb{R}^4\setminus B_\rho (0)$ et équivalente de jauge à un instanton dans une boule plus petite $B_{\tau \rho }(0)$, et de telle sorte que le coût d’énergie de Yang–Mills soit strictement inférieur à la somme de $\operatorname{YM}(A)$ et de l’énergie de l’instanton inséré. Précisément nous avons

\[ \operatorname{YM}(\hat{A})\le \operatorname{YM}(A)+8\pi ^2-\varepsilon _0\rho ^4\left|F_A(0) \right|^2, \]

où $\tau \in (0.3,0.4)$ et $\varepsilon _0>0$ sont des constantes universelles indépendantes de $A$ et de $\rho $. Notre méthode de recollement est similaire dans l’esprit à celle de Brezis–Coron pour les applications harmoniques utilisée pour construire de nouvelles solutions. Nous la comparons par ailleurs avec des résultats classiques de Taubes sur les champs de Yang–Mills et discutons ses applications existantes et potentielles ainsi que des problèmes ouverts.

Reçu le : 2025-05-21
Révisé le : 2025-09-09
Accepté le : 2025-09-16
Publié le : 2025-10-13

DOI : 10.5802/crmath.795

Affiliations des auteurs :

Luca Martinazzi ¹ ; Tristan Rivière ²

¹ Department of Mathematics Guido Castelnuovo, Università di Roma La Sapienza, Italy
² Department of Mathematics, ETH Zentrum, 8093 Zürich, Switzerland

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Luca Martinazzi and Tristan Rivi\`ere},
     title = {Gluing instantons \`a la {Brezis{\textendash}Coron} in dimension four and the dipole construction},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1219--1261},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.795},
     language = {en},
}

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Luca Martinazzi; Tristan Rivière. Gluing instantons à la Brezis–Coron in dimension four and the dipole construction. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1219-1261. doi: 10.5802/crmath.795

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