Details
| Original language | English |
|---|---|
| Pages (from-to) | 97-121 |
| Number of pages | 25 |
| Journal | Journal fur die Reine und Angewandte Mathematik |
| Volume | 2019 |
| Issue number | 750 |
| Early online date | 30 Aug 2016 |
| Publication status | Published - 2019 |
Abstract
In [18], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifold is related to the Lagrangian angle defined by the phase of a parallel (n, 0)-form, just like the Calabi-Yau case. We also show that the corresponding Lagrangian mean curvature flow in cotangent bundles preserves the exactness and the zero Maslov class conditions. At the end, we prove a long time existence and convergence result to demonstrate the stability of the zero section of the cotangent bundle of spheres.
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
- Mathematics(all)
- Applied Mathematics
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In: Journal fur die Reine und Angewandte Mathematik, Vol. 2019, No. 750, 2019, p. 97-121.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Generalized Lagrangian mean curvature flows: The cotangent bundle case
AU - Smoczyk, Knut
AU - Tsui, Mao Pei
AU - Wang, Mu Tao
N1 - Funding information: The first named author was supported by the DFG (German Research Foundation). The second named author was partially supported by a Collaboration Grant for Mathematicians from the Simons Foundation #239677 and in part by Taiwan MOST grant 104-2115-M-002-001-MY2. This material is based upon work supported by the National Science Foundation under Grant Numbers DMS-1105483 and DMS-1405152 (Mu-Tao Wang).
PY - 2019
Y1 - 2019
N2 - In [18], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifold is related to the Lagrangian angle defined by the phase of a parallel (n, 0)-form, just like the Calabi-Yau case. We also show that the corresponding Lagrangian mean curvature flow in cotangent bundles preserves the exactness and the zero Maslov class conditions. At the end, we prove a long time existence and convergence result to demonstrate the stability of the zero section of the cotangent bundle of spheres.
AB - In [18], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifold is related to the Lagrangian angle defined by the phase of a parallel (n, 0)-form, just like the Calabi-Yau case. We also show that the corresponding Lagrangian mean curvature flow in cotangent bundles preserves the exactness and the zero Maslov class conditions. At the end, we prove a long time existence and convergence result to demonstrate the stability of the zero section of the cotangent bundle of spheres.
UR - http://www.scopus.com/inward/record.url?scp=85065064264&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1604.02936
DO - 10.48550/arXiv.1604.02936
M3 - Article
AN - SCOPUS:85065064264
VL - 2019
SP - 97
EP - 121
JO - Journal fur die Reine und Angewandte Mathematik
JF - Journal fur die Reine und Angewandte Mathematik
SN - 0075-4102
IS - 750
ER -