### Abstract

In this paper, we generalize several results for the Hamiltonian stability and the mean curvature flow of Lagrangian submanifolds in a Kähler–Einstein manifold to more general Kähler manifolds including a Fano manifold equipped with a Kähler form ω∈2πc_{1}(M) by using the method proposed by Behrndt (2011). Namely, we first consider a weighted measure on a Lagrangian submanifold L in a Kähler manifold M and investigate the variational problem of L for the weighted volume functional. We call a stationary point of the weighted volume functional f-minimal, and define the notion of Hamiltonian f-stability as a local minimizer under Hamiltonian deformations. We show such examples naturally appear in a toric Fano manifold. Moreover, we consider the generalized Lagrangian mean curvature flow in a Fano manifold which is introduced by Behrndt and Smoczyk–Wang. We generalize the result of H. Li, and show that if the initial Lagrangian submanifold is a small Hamiltonian deformation of an f-minimal and Hamiltonian f-stable Lagrangian submanifold, then the generalized MCF converges exponentially fast to an f-minimal Lagrangian submanifold.

Original language | English |
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Pages (from-to) | 140-168 |

Number of pages | 29 |

Journal | Journal of Geometry and Physics |

Volume | 128 |

DOIs | |

Publication status | Published - 2018 Jun |

### Keywords

- Generalized Lagrangian mean curvature flow
- Hamiltonian stability
- Lagrangian submanifolds
- Toric Fano manifolds

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Geometry and Topology