TY - JOUR

T1 - Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow

AU - Kajigaya, Toru

AU - Kunikawa, Keita

N1 - Funding Information:
The authors would like to thank Shouhei Honda and Hikaru Yamamoto for helpful comments. A part of this work was done while the first author was staying at University of Tübingen by the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, Mathematical Science of Symmetry, Topology and Moduli, Evolution of International Research Network based on OCAMI. He is grateful for the hospitality of the University. The second author is supported by the Grant-in-Aid for JSPS Fellows 16J01498 . During the preparation of this paper he has stayed in Max Planck Institute for Mathematics in the Sciences, Leipzig. He is grateful to Jürgen Jost for the hospitality.

PY - 2018/6

Y1 - 2018/6

N2 - 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∈2πc1(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.

AB - 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∈2πc1(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.

KW - Generalized Lagrangian mean curvature flow

KW - Hamiltonian stability

KW - Lagrangian submanifolds

KW - Toric Fano manifolds

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U2 - 10.1016/j.geomphys.2018.02.011

DO - 10.1016/j.geomphys.2018.02.011

M3 - Article

AN - SCOPUS:85043373340

VL - 128

SP - 140

EP - 168

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

ER -