A Hamiltonian Stable Minimal Lagrangian Submanifold of Projective Space with Nonparallel Second Fundamental Form

Abstract

In this paper we show that Hamiltonian stable minimal Lagrangian submanifolds of projective space need not have parallel second fundamental form.

On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres

“…Recently the above result of [1] is generalized as follows ( [2,3]): If L is a compact parallel Lagrangian submanifold embedded in a complex space form (=CP n , C n or CH n ), then L is Hamiltonian stable. More recently, an example of a compact Hamiltonian stable minimal Lagrangian submanifold in CP 3 with ∇S = 0, which is obtained as a minimal Lagrangian SU(2)-orbit in CP 3 , was shown by L. Bedulli and A. Gori ( [6]), and independently by [33]: (e) The orbit ρ 3 (SU(2))[z 3 0 + z 3 1 ] ⊂ CP 3 by the irreducible unitary representation of SU(2) of degree 3 (cf. Section 5) is a 3-dimensional compact embedded Hamiltonian stable minimal Lagrangian submanifold with ∇S = 0.…”

Geometry of Certain Lagrangian Submanifolds in Hermitian Symmetric Spaces

“…The simplest non-trivial example can be given by a 3dimensional minimal Lagrangian orbit in CP 2 under an irreducible unitary representation of SU (2) of degree 3 (cf. [3], [13]).…”

Construction of homogeneous Lagrangian submanifolds in $\boldsymbol{CP}^n$ and Hamiltonian stability

“…Let {e 1 , e 2 } the standard basis of C 2 . We may define a unitary structure on S 2 (C 2 ) with orthonormal basis given by {e 2 1 , √ 2e 1 e 2 , e 2 2 } with respect to which the induced action of SU(2) becomes unitary. By tensoring with the standard basis of C 2 we get an orthonormal basis of V .…”

“…If we endow CP n with the standard Fubini-Study metric g FS with holomorphic sectional curvature c, then Oh ( [13]) proved that a minimal Lagrangian submanifold L is stable if and only if the first eigenvalue λ 1 (L) for the Laplacian ∆ relative to the induced metric and acting on C ∞ (L) satisfies λ 1 (L) ≥ n+1 2 c. Actually, since λ 1 (L) ≤ n+1 2 c for every minimal Lagrangian submanifold of CP n by a result due to Ono ([15]), we see that stability is equivalent to λ 1 (L) = n+1 2 c. It is a natural and interesting problem to classify all minimal, Hamiltonian stable Lagrangian submanifolds of CP n . In [1], Amarzaya and Ohnita prove that every minimal Lagrangian submanifold with parallel second fundamental form is actually stable, while Bedulli and Gori ( [2]) and independently Ohnita ([14]) exhibited the first example of a Hamiltonian stable Lagrangian submanifold which has non-parallel second fundamental form. This example sits inside CP 3 and is homogeneous under the action of the group SU(2).…”