[Paper Review] Lagrangian submanifolds with prescribed second fundamental form

Bang‐Yen Chen, Joeri Van der Veken+1 more|arXiv (Cornell University)|Sep 17, 2013

Geometric Analysis and Curvature Flows4 references4 citations

TL;DR

This paper classifies Lagrangian submanifolds in complex space forms (ℂⁿ, ℂPⁿ(4), ℂHⁿ(−4)) whose second fundamental form exhibits a specific tensorial structure depending on a real parameter d. Using a warped product structure and solving a system of ODEs for curvature functions λ, μ, θ, the authors construct explicit immersions via minimal Legendrian submanifolds, identifying cases where equality is achieved in delta-curvature inequalities—particularly for δ(m,…,m) and δ(2,n−2) ideals when d = 1/(2+m) or d = 1/(n−1).

ABSTRACT

We classify Lagrangian submanifolds of complex space forms, whose second fundamental form can be written in a certain way, depending on a real parameter. For some special values of this parameter, the resulting submanifolds are ideal in the sense that they realize equality in an inequality for a Chen's delta-curvature.

Motivation & Objective

• To classify Lagrangian submanifolds in complex space forms with a second fundamental form that satisfies a specific tensorial structure parameterized by a real constant d.
• To identify conditions under which such submanifolds achieve equality in delta-curvature inequalities, thus realizing ideal submanifolds.
• To construct explicit local immersions of these submanifolds in ℂⁿ, ℂPⁿ(4), and ℂHⁿ(−4) using solutions to a system of ordinary differential equations.
• To determine the geometric structure of the submanifolds, showing they are intrinsically warped products I ×_f N with e₁ tangent to the interval I and e₂,…,eₙ tangent to N.
• To clarify the role of special d-values (e.g., d = 1/(n+1), d = 1/(n−1)) in realizing δ(n−1)-ideal or δ(2,n−2)-ideal submanifolds.

Proposed method

• Assumes the existence of a local orthonormal frame {e₁,…,eₙ} such that the second fundamental form h takes a specific form involving λ, d, and J, with h(e₁,e₁) = λJe₁ and off-diagonal terms involving dλJeᵢ.
• Derives a system of ODEs for functions λ(t), μ(t), θ(t) along the integral curves of e₁, with λ′ = ((1−2d)/d)λμ, μ′ = ±(1 + μ² + d(1−d)λ²) or similar depending on ambient curvature.
• Uses the horizontal lift of the immersion to the sphere S²ⁿ⁺¹(1) or hyperbolic space H²ⁿ⁺¹₁(−1), depending on the ambient space, to express the full immersion in terms of φ (a minimal Legendrian immersion) and the ODE solutions.
• Applies the warped product structure I ×_f N, where I is an interval parameterized by t, and N is an (n−1)-dimensional manifold with a minimal Lagrangian immersion φ.
• Solves the ODE system to determine the coefficient functions μ(t), θ(t), and λ(t), which control the phase and scaling in the immersion formula.
• Distinguishes cases based on the sign of 1 − μ² − d²λ² in the hyperbolic case, leading to different parametrizations of the horizontal lift.

Experimental results

Research questions

• RQ1 For which values of the parameter d do Lagrangian submanifolds with the specified second fundamental form become δ-ideal?
• RQ2 How can one explicitly construct Lagrangian submanifolds in complex space forms with such a second fundamental form?
• RQ3 What is the intrinsic geometric structure of these submanifolds, and how does it relate to warped products and minimal Legendrian submanifolds?
• RQ4 What role do the ODEs for λ(t), μ(t), and θ(t) play in determining the global and local geometry of the immersion?
• RQ5 How do special values of d (e.g., d = 1/(n+1), d = 1/(n−1)) correspond to known classes of ideal submanifolds in curvature inequalities?

Key findings

• The submanifolds are intrinsically warped products I ×_f N, with e₁ tangent to I and e₂,…,eₙ tangent to N, and the warping function f determined by the solution of the ODE system.
• For d = 1/(2+m) with m ≥ 2 dividing n−1, the submanifolds are δ(m,…,m)-ideal (k times m, km = n−1), achieving equality in the corresponding delta-curvature inequality.
• For d = 1/(n−1) and n ≥ 5, under an additional orthogonality condition, the submanifolds are δ(2,n−2)-ideal, realizing equality in that inequality.
• The immersion in ℂⁿ is given by L(t,u₂,…,uₙ) = (e^{idθ}ϕ / √(1+μ²+d²λ²), e^{i(1−d)θ}(idλ−μ)/√(1+μ²+d²λ²)), with ϕ a minimal Legendrian immersion into S²ⁿ⁻¹(1).
• In ℂPⁿ(4), the horizontal lift is expressed similarly, with the same ODE system and ϕ minimal in S²ⁿ⁻¹(1), up to isometries.
• In ℂHⁿ(−4), the immersion splits into three cases based on the sign of 1−μ²−d²λ², with distinct parametrizations involving √(1−μ²−d²λ²) or √(μ²+d²λ²−1), and ϕ minimal in H²ⁿ⁻¹₁(−1) or S²ⁿ⁻¹(1) respectively.