[Paper Review] Sasaki manifolds, Kaehler cone manifolds and biharmonic submanifolds

Hajime Urakawa|arXiv (Cornell University)|Jun 26, 2013

Geometry and complex manifolds14 references3 citations

TL;DR

This paper establishes a correspondence between biharmonic Legendrian submanifolds in Sasaki manifolds and biharmonic Lagrangian cones in Kähler cone manifolds. It proves that the cone submanifold is biharmonic if and only if it is harmonic, and shows that the original submanifold is proper biharmonic precisely when its tension field is a non-zero eigen-section of the Jacobi operator with eigenvalue equal to the dimension of the submanifold.

ABSTRACT

For a Legendrian submanifold $M$ of a Sasaki manifold $N$, we study harmonicity and biharmonicity of the corresponding Lagrangian cone submanifold C(M) of a Kaehler manifold C(N). We show that, if $C(M)$ is biharmonic in C(N), then it is harmonic; and $M$ is proper biharmonic in $N$ if and only if C(M) has a non-zero eigen-section of the Jacobi operator with the eigenvalue $m=dim M$.

Motivation & Objective

• To investigate the relationship between biharmonicity of Legendrian submanifolds in Sasaki manifolds and their associated Lagrangian cones in Kähler cone manifolds.
• To determine when the cone construction preserves biharmonicity and relate it to harmonicity.
• To characterize proper biharmonic Legendrian submanifolds via spectral properties of the Jacobi operator on the cone.
• To extend Takahashi-type eigenvalue theorems to the setting of biharmonic maps and Sasaki geometry.

Proposed method

• Utilizes the cone construction that maps a Legendrian submanifold $M^m$ in a Sasaki manifold $N^{2m+1}$ to a Lagrangian submanifold $C(M)$ in the Kähler cone $C(N)$.
• Applies the first and second variation formulas for the bienergy functional to derive the biharmonicity equations for the cone map $\overline{\varphi}$.
• Employs the Jacobi operator $J_{\overline{\varphi}}$ on the pullback bundle of the cone map to analyze eigen-sections of the tension field.
• Uses the curvature structure of Sasaki space forms with constant $J$-sectional curvature $\epsilon$ to simplify the biharmonicity equations.
• Applies Takahashi's theorem on eigenfunctions of the Laplacian to reinterpret the eigenvalue condition in terms of the dimension $m$ of the submanifold.
• Establishes equivalence between proper biharmonicity of $\varphi$ and the tension field $\tau(\overline{\varphi})$ being a non-zero eigen-section of the rough Laplacian with eigenvalue $m$.

Experimental results

Research questions

• RQ1 Under what conditions is the Lagrangian cone submanifold $C(M)$ of a Legendrian submanifold $M$ in a Sasaki manifold $N$ biharmonic?
• RQ2 When is a Legendrian submanifold $M$ in a Sasaki manifold $N$ properly biharmonic?
• RQ3 How does the biharmonicity of the cone map $\overline{\varphi}$ relate to the harmonicity of the original map $\varphi$?
• RQ4 What spectral condition on the Jacobi operator characterizes proper biharmonic Legendrian submanifolds in the standard sphere $S^{2m+1}(1)$?
• RQ5 Can the Takahashi-type eigenvalue condition for harmonic maps be generalized to the biharmonic setting via the cone construction?

Key findings

• The cone submanifold $C(M)$ is biharmonic in $C(N)$ if and only if it is harmonic, which occurs precisely when the original Legendrian submanifold $M$ is harmonic in $N$.
• A Legendrian submanifold $\varphi: M^m \to N^{2m+1}$ is proper biharmonic if and only if its tension field $\tau(\overline{\varphi})$ is a non-zero eigen-section of the rough Laplacian $\overline{\overline{\Delta}}_{\overline{\varphi}}$ with eigenvalue $m = \dim M$.
• In the case of the standard sphere $S^{2m+1}(1)$, the cone map $\overline{\varphi}: C(M) \to \mathbb{C}^{m+1}$ satisfies $\overline{\overline{\Delta}}_{\overline{\varphi}} \tau(\overline{\varphi}) = m \tau(\overline{\varphi})$ if and only if $\varphi$ is proper biharmonic.
• The biharmonicity of $\varphi$ is equivalent to the vanishing of two components: the rough Laplacian of the mean curvature field and a curvature-corrected second fundamental form term.
• For Sasaki space forms with constant $J$-sectional curvature $\epsilon$, the biharmonicity condition reduces to $\overline{\Delta}_{\varphi} \mathbf{H} = \frac{\epsilon(m+3) + 3(m-1)}{4} \mathbf{H}$, linking the geometry of $N$ to the spectral properties of $\mathbf{H}$.
• The result generalizes Takahashi's theorem to biharmonic maps: just as coordinate functions of minimal immersions into spheres are eigenfunctions of the Laplacian with eigenvalue $m$, the tension field of the cone map is an eigen-section of the rough Laplacian with the same eigenvalue $m$.