, En particulier, les sous-variétés complexes sont minimales

, On dispose maintenant de toutes les notions nécessaires à la démonstration du théorème 6. Récapitulons la construction menée au chapitre 3

C. Dans-ce,

, sur des variétés symplectiques lisses (M ? , ? ? ), de sorte que J f soit compatible avec ? ? au sens de la section 1.2.1, et que la structure presque-kählérienne résultante (? ?

, De plus, lorsque ? tend vers 0, la solution (J f , g f ) converge vers (J X , g X ) sur un voisinage compact de la section nulle dans X T * S 2 , au sens de 3.2.2. On a, en effet, les estimées suivantes : h * ?, J f ? J X C, vol.2

, 1, les variétés (M ? , ? ? ) sont symplectomorphes et s'identifient donc toutes à une même variété symplectique que l'on notera ( ? M , ? ?)

, on note alors J ? et g ? les tirés en arrière de J f et g f sur?Msur? sur?M. On note aussi (J 0 , g 0 ) le tiré en arrière de la solution approchée

, On dispose alors d'une famille lisse (J ? , g ? ) 0??<?0 de structures presque-kählériennes sur une variété symplectique fixée

. Observons-maintenant-que and . Dans-le-modèle-ale-(x-t-*-s-2, la section nulle S 0 de T * S 2 ? S 2 est une sphère lagrangienne, puisque la forme de Kähler est exacte. De plus, comme on l'a vu au chapitre 2, section 2.2.2, T * S 2 peut être munie d'une structure hyperKähler compatible avec la métrique d'Eguchi-Hanson g X. Pour un choix différent de structure complexe dans la famille hyperKähler

, 2, la section nulle minimise donc le volume dans sa classe d'homologie

, La section nulle n'est pas holomorphe pour notre choix de structure complexe sur T * S 2. Cependant, puisque la métrique est la même (c'est la métrique d'Eguchi-Hanson), S 0 est encore minimale ; en particulier, elle est hamiltonienne stationnaire

?. , 0 fournit une sphère S hamiltonienne stationnaire (puisque minimale) dans la 'somme connexe

C. Stationnaires,

. Or and . Le-théorème-5,-j-?-est-lisse, et il en est de même de la métrique associée g ? dont les coefficients apparaissent dans l'expression de l'opérateur différentiel B. Une fois encore, on peut donc appliquer un argument de bootstrapping pour s'assurer que les solutions

, On peut alors s'interroger sur la possibilité d'étendre la seconde partie du résultat de Biquard et Rollin [17], Theorem D-c'est-à-dire

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