LAGRANGIAN SUBMANIFOLDS IN STRICT NEARLY KӒHLER 6-MANIFOLDS

Lê, Hông Vân; Schwachhöfer, Lorenz

doi:info:doi/10.18910/73359

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タイトル	LAGRANGIAN SUBMANIFOLDS IN STRICT NEARLY KӒHLER 6-MANIFOLDS
著者	Lê, Hông Vân Lê, Hông Vân
著者	Schwachhöfer, Lorenz Schwachhöfer, Lorenz
抄録	Lagrangian submanifolds in strict nearly Kähler 6-manifolds are related to special Lagrangian submanifolds in Calabi-Yau 6-manifolds and coassociative cones in 𝐺_2-manifolds. We prove that the mean curvature of a Lagrangian submanifold 𝐿 in a nearly Kähler manifold (𝑀, 𝐽, 𝑔) is symplectically dual to the Maslov 1-form on 𝐿. Using relative calibrations, we derive a formula for the second variation of the volume of a Lagrangian submanifold 𝐿³ in a strict nearly Kähler manifold (𝑀^6, 𝐽, 𝑔) and compare it with McLean’s formula for special Lagrangian submanifolds. We describe a finite dimensional local model of the moduli space of compact Lagrangian submanifolds in a strict nearly Kähler 6-manifold. We show that there is a real analytic atlas on (𝑀^6, 𝐽, 𝑔) in which the strict nearly Kähler structure (𝐽, 𝑔) is real analytic. Furthermore, w.r.t. an analytic strict nearly Kähler structure the moduli space of Lagrangian submanifolds of 𝑀^6 is a real analytic variety, whence infinitesimal Lagrangian deformations are smoothly obstructed if and only if they are formally obstructed. As an application, we relate our results to the description of Lagrangian submanifolds in the sphere 𝑆^6 with the standard nearly Kähler structure described in [34].
公開者	Osaka University and Osaka City University, Departments of Mathematics
掲載誌名	Osaka Journal of Mathematics
巻	56
号	3
開始ページ	601
終了ページ	629
刊行年月	2019-07
ISSN	00306126
NCID	AA00765910
URL	http://hdl.handle.net/11094/73359
言語	英語
DOI	info:doi/10.18910/73359
カテゴリ	紀要論文 Departmental Bulletin Paper
カテゴリ	Osaka Journal of Mathematics / Volume 56, Number 3 （2019-7）

著者版フラグ	publisher
NII資源タイプ	紀要論文
ローカル資源タイプ	紀要論文
dcmi資源タイプ	text
DCTERMS.bibliographicCitation	Osaka Journal of Mathematics.56(3) P.601-P.629
DC.title	LAGRANGIAN SUBMANIFOLDS IN STRICT NEARLY KӒHLER 6-MANIFOLDS
DC.creator	Lê, Hông Vân
DC.creator	Schwachhöfer, Lorenz
DC.publisher	Osaka University and Osaka City University, Departments of Mathematics
DC.language" scheme="DCTERMS.RFC1766	英語
DCTERMS.issued" scheme="DCTERMS.W3CDTF	2019-07
DC.identifier" scheme="DCTERMS.URI	http://hdl.handle.net/11094/73359
DCTERMS.abstract	Lagrangian submanifolds in strict nearly Kähler 6-manifolds are related to special Lagrangian submanifolds in Calabi-Yau 6-manifolds and coassociative cones in 𝐺_2-manifolds. We prove that the mean curvature of a Lagrangian submanifold 𝐿 in a nearly Kähler manifold (𝑀, 𝐽, 𝑔) is symplectically dual to the Maslov 1-form on 𝐿. Using relative calibrations, we derive a formula for the second variation of the volume of a Lagrangian submanifold 𝐿³ in a strict nearly Kähler manifold (𝑀^6, 𝐽, 𝑔) and compare it with McLean’s formula for special Lagrangian submanifolds. We describe a finite dimensional local model of the moduli space of compact Lagrangian submanifolds in a strict nearly Kähler 6-manifold. We show that there is a real analytic atlas on (𝑀^6, 𝐽, 𝑔) in which the strict nearly Kähler structure (𝐽, 𝑔) is real analytic. Furthermore, w.r.t. an analytic strict nearly Kähler structure the moduli space of Lagrangian submanifolds of 𝑀^6 is a real analytic variety, whence infinitesimal Lagrangian deformations are smoothly obstructed if and only if they are formally obstructed. As an application, we relate our results to the description of Lagrangian submanifolds in the sphere 𝑆^6 with the standard nearly Kähler structure described in [34].
DC.identifier	info:doi/10.18910/73359
citation_title	LAGRANGIAN SUBMANIFOLDS IN STRICT NEARLY KӒHLER 6-MANIFOLDS
citation_author	Lê, Hông Vân
citation_author	Schwachhöfer, Lorenz
citation_publisher	Osaka University and Osaka City University, Departments of Mathematics
citation_language	英語
citation_date	2019-07
citation_journal_title	Osaka Journal of Mathematics
citation_volume	56
citation_issue	3
citation_firstpage	601
citation_lastpage	629
citation_issn	00306126
citation_public_url	http://hdl.handle.net/11094/73359
citation_doi	info:doi/10.18910/73359

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