Second variation formula and the stability of Legendrian minimal submanifolds in Sasakian manifolds

Abstract

In this paper, we investigate compact Legendrian submanifolds L in Sasakian manifolds M, which have extremal volume under Legendrian deformations. We call such a submanifold L-minimal Legendrian submanifold. We derive the second variational formula for the volume of L under Legendrian deformations in M. Applying this formula, we investigate the stability of L-minimal Legendrian curves in Sasakian space forms, and show the L-instability of L-minimal Legendrian submanifolds in S 2n+1 (1). Moreover, we give a construction of L-minimal Legendrian submanifolds in R 2n+1 (−3). 1. Introduction. In [10], [11], Y. G. Oh introduced the notion of Hamiltonianminimal (H-minimal) Lagrangian submanifolds in Kähler manifolds. Such a submanifold is a critical point of the volume functional under the Hamiltonian deformation. This is an extension of the notion of minimal submanifold, and has been studied by many authors (for example, [7], [8], [11], [14] and see references therein). An H-minimal Lagrangian submanifold is called Hamiltonian-stable (H-stable) if the second variation is non-negative for any Hamiltonian deformation. Oh studied H-stablity of some examples of H-minimal Lagrangian submanifold in a specific Kähler manifold ([10], [11]). For example, the real projective space RP n and the Clifford torus in CP n , and the standard tori in C n are H-stable. Besides these examples, Schoen and Wolfson studied the H-stablity of two-dimensional H-minimal Lagrangian cones ([14]), and Iriyeh studied the three-dimensional case ([8]). Furthermore, in [1], Amarzaya and Ohnita proved that all compact Lagrangian submanifolds with parallel second fundamental form in C n and CP n are H-stable.On the other hand, there is a notion of Sasakian manifolds, which is an odd-dimensional counterpart to Kähler manifolds. In Sasakian manifolds, we consider Legendrian-minimal (Lminimal) Legendrian submanifolds which correspond to H-minimal Lagrangian manifolds in Kähler manifolds. An L-minimal Legendrian submanifold is a critical point of the volume function under the Legendrian deformation (for more details, see Section 3). In [7], [8], the authors constructed examples of L-minimal Legendrian submanifolds in odd-dimensional unit spheres (in [7], such submanifolds are called C-minimal). Moreover, they show that a certain L-minimal Legendrian submanifold in a Sasakian manifold is related to an H-minimal Lagrangian submanifold in a Kähler manifold. For example, the cone over a Legendrian

Second variation for L-minimal Legendrian submanifolds in pseudo-Sasakian manifolds

“…where we used that X vanishes on the boundary. Since this vanishes for arbitrary Legendrian variations we conclude (13).…”

F-biharmonic maps into general Riemannian manifolds

“…For harmonic maps, it is well know that: If a domain manifold (M, g) is complete and has non-negative Ricci curvature and the sectional curvature of a target manifold (N, h) is non-positive, then every nite harmonic map is a constant map [27]. See [28] and [29] for recent works on harmonic map. If M is noncompact, the maximum principle is no longer application.…”

Magnetic curves in the real special linear group

“…To look for periodic trajectories, we restrict our attention to horizontal lifts of closed Riemannian circles. See also [21,Example 5.5].…”