Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow

Reductions of minimal Lagrangian submanifolds with symmetries

“…We remark that the closed 1-formsαH andβH HL are referred as generalized mean curvature forms in [5] and [27] or Maslov forms in [21] of φ and φ 0 , respectively. Namely,αH (orβH HL ) is regarded as a "connection form" of a unitary connection ∇ on the trivial bundle φ * K M in the trivialization Ω L defined by a unique extension of the volume form of φ:∇ [27] or [18]), and we use the same symbol for the induced connection on φ * K M . This can be easily shown by using Proposition 4.2 in [21].…”

“…This can be easily shown by using Proposition 4.2 in [21]. As shown in [5], [18], [21] and [27], there are several advantages to consider the (closed) Maslov forms in the non Kähler-Einstein setting. This point is a crucial difference betweenαH and α H in our setting, and gives a reason why we considerαH instead of α H .…”

“…any Hamiltonian deformation of L in the sense of [24]). Minimal Lagrangian submanifolds are of particular interests in several contexts, e.g., the Lagrangian mean curvature flows and the Hamiltonian volume minimizing problem (see [18], [21], [24], [27] and references therein).When M is a Kähler manifold, the symplectic quotient space M c inherits a natural Kähler structure, and we call this reduction procedure the Kähler reduction (see Section 2). In [11], Dong applied Hsiang-Lawson's method to Hamiltonian minimal Lagrangian submanifolds in a Kähler manifold M and proved that a Kinvariant Lagrangian submanifold L in M is Hamiltonian minimal with respect to the Kähler metric g if and only if so is L c in the Kähler quotient M c with respect to the Hsiang-Lawson metric of g. By using this reduction method, Dong constructed infinitely many Hamiltonian minimal Lagrangian submanifolds with large symmetries in CP n and C n .…”

“…any Hamiltonian deformation of L in the sense of [24]). Minimal Lagrangian submanifolds are of particular interests in several contexts, e.g., the Lagrangian mean curvature flows and the Hamiltonian volume minimizing problem (see [18], [21], [24], [27] and references therein).…”

On Hamiltonian stable Lagrangian tori in complex hyperbolic spaces

“…We refer to [1], [13], [16], [17], [18] and references therein for explicit examples of H-stable homogeneous Lagrangians in a Hermitian symmetric space, and [10] for existence of H-stable Lagrangians in a general compact almost Kähler manifold. See also [12] for a generalization of the notion of H-stability.…”