Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 129-140 |
| Seitenumfang | 12 |
| Fachzeitschrift | Asian Journal of Mathematics |
| Jahrgang | 15 |
| Ausgabenummer | 1 |
| Publikationsstatus | Veröffentlicht - März 2011 |
Abstract
An almost Kähler structure on a symplectic manifold (N, ω) consists of a Riemannian metric g and an almost complex structure J such that the symplectic form ω satisfies ω(·, ·) = g(J(·), ·). Any symplectic manifold admits an almost Kähler structure and we refer to (N, ω, g, J) as an almost Kähler manifold. In this article, we propose a natural evolution equation to investigate the deformation of Lagrangian submanifolds in almost Kähler manifolds. A metric and complex connection ∇̂ on TN defines a generalized mean curvature vector field along any Lagrangian submanifold M of N. We study the evolution of M along this vector field, which turns out to be a Lagrangian deformation, as long as the connection ∇̂ satisfies an Einstein condition. This can be viewed as a generalization of the classical Lagrangian mean curvature flow in Kähler-Einstein manifolds where the connection ∇̂ is the Levi-Civita connection of g. Our result applies to the important case of Lagrangian submanifolds in a cotangent bundle equipped with the canonical almost Kähler structure and to other generalization of Lagrangian mean curvature flows, such as the flow considered by Behrndt [B] in Kähler manifolds that are almost Einstein.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Allgemeine Mathematik
- Mathematik (insg.)
- Angewandte Mathematik
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in: Asian Journal of Mathematics, Jahrgang 15, Nr. 1, 03.2011, S. 129-140.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Generalized Lagrangian mean curvature flows in symplectic manifolds
AU - Smoczyk, Knut
AU - Wang, Mu Tao
PY - 2011/3
Y1 - 2011/3
N2 - An almost Kähler structure on a symplectic manifold (N, ω) consists of a Riemannian metric g and an almost complex structure J such that the symplectic form ω satisfies ω(·, ·) = g(J(·), ·). Any symplectic manifold admits an almost Kähler structure and we refer to (N, ω, g, J) as an almost Kähler manifold. In this article, we propose a natural evolution equation to investigate the deformation of Lagrangian submanifolds in almost Kähler manifolds. A metric and complex connection ∇̂ on TN defines a generalized mean curvature vector field along any Lagrangian submanifold M of N. We study the evolution of M along this vector field, which turns out to be a Lagrangian deformation, as long as the connection ∇̂ satisfies an Einstein condition. This can be viewed as a generalization of the classical Lagrangian mean curvature flow in Kähler-Einstein manifolds where the connection ∇̂ is the Levi-Civita connection of g. Our result applies to the important case of Lagrangian submanifolds in a cotangent bundle equipped with the canonical almost Kähler structure and to other generalization of Lagrangian mean curvature flows, such as the flow considered by Behrndt [B] in Kähler manifolds that are almost Einstein.
AB - An almost Kähler structure on a symplectic manifold (N, ω) consists of a Riemannian metric g and an almost complex structure J such that the symplectic form ω satisfies ω(·, ·) = g(J(·), ·). Any symplectic manifold admits an almost Kähler structure and we refer to (N, ω, g, J) as an almost Kähler manifold. In this article, we propose a natural evolution equation to investigate the deformation of Lagrangian submanifolds in almost Kähler manifolds. A metric and complex connection ∇̂ on TN defines a generalized mean curvature vector field along any Lagrangian submanifold M of N. We study the evolution of M along this vector field, which turns out to be a Lagrangian deformation, as long as the connection ∇̂ satisfies an Einstein condition. This can be viewed as a generalization of the classical Lagrangian mean curvature flow in Kähler-Einstein manifolds where the connection ∇̂ is the Levi-Civita connection of g. Our result applies to the important case of Lagrangian submanifolds in a cotangent bundle equipped with the canonical almost Kähler structure and to other generalization of Lagrangian mean curvature flows, such as the flow considered by Behrndt [B] in Kähler manifolds that are almost Einstein.
KW - Almost kähler structure
KW - Cotangent bundle
KW - Lagrangian mean curvature flow
KW - Symplectic manifold
UR - http://www.scopus.com/inward/record.url?scp=79954463978&partnerID=8YFLogxK
U2 - 10.4310/AJM.2011.v15.n1.a7
DO - 10.4310/AJM.2011.v15.n1.a7
M3 - Article
AN - SCOPUS:79954463978
VL - 15
SP - 129
EP - 140
JO - Asian Journal of Mathematics
JF - Asian Journal of Mathematics
SN - 1093-6106
IS - 1
ER -