[Paper Review] Holomorphic Cubic Differentials and Minimal Lagrangian Surfaces in CH2

Zheng Huang, John Loftin+1 more|arXiv (Cornell University)|Jan 18, 2012

Geometric Analysis and Curvature Flows10 references3 citations

TL;DR

This paper establishes the existence and non-uniqueness of minimal Lagrangian immersions of the universal cover of closed surfaces (genus ≥ 2) into complex hyperbolic space ℂH², parameterized by a conformal structure σ and a holomorphic cubic differential tq. It shows that the surface area of such immersions, as a functional on the space of solutions, serves as a Weil-Petersson potential for the space of holomorphic cubic differentials, with second variation equal to 16 times the Weil-Petersson norm of q.

ABSTRACT

Following earlier work of Loftin-McIntosh, we study minimal Lagrangian immersions of the universal cover of a closed surface (of genus at least 2) into CH2, with prescribed data of a conformal structure plus a holomorphic cubic differential. We show existence and non-uniqueness of such minimal Lagrangian immersions. We also establish the surface area with respect to the induced metric as a Weil-Petersson potential function for the space of holomorphic cubic differentials on the Riemann surface.

Motivation & Objective

• To establish the existence and non-uniqueness of minimal Lagrangian immersions of the universal cover of closed surfaces (genus g ≥ 2) into ℂH².
• To analyze the asymptotic behavior of such immersions as the parameter t → 0.
• To show that the surface area of the induced metric e^u g_σ is a Weil-Petersson potential function for the space of holomorphic cubic differentials on (Σ, σ).
• To develop a moduli theory for minimal Lagrangian immersions into ℂH² using the solution curve of a nonlinear PDE.

Proposed method

• Reduces the minimal Lagrangian immersion problem to solving a nonlinear PDE: Δu + 2 - 2e^u - 16t²‖q‖²e^{-2u} = 0 on a compact Riemann surface with hyperbolic metric g_σ.
• Uses the solution u(t) to construct a Legendrian frame from the universal cover ˜Σ to SU(2,1), yielding an equivariant minimal Lagrangian immersion φ: ˜Σ → ℂH².
• Applies the implicit function theorem and mountain-pass methods to prove existence and non-uniqueness of solutions for small t > 0.
• Utilizes the Weil-Petersson pairing ⟨q₁, q₂⟩_WP = ∫_Σ (q₁ ̄q₂)/g_σ³ dA_σ to relate the second variation of the area functional to the geometry of Teichmüller space.
• Analyzes the behavior of solutions along a continuous curve γ(t) = (u(t), t) in the solution space, showing that u(t) → 0 as t → 0.
• Applies the maximum principle and H¹-estimates to rule out blow-up of solutions and establish convergence of sequences of solutions.

Experimental results

Research questions

• RQ1 Does there exist a minimal Lagrangian immersion of the universal cover of a closed surface of genus g ≥ 2 into ℂH² for any given conformal structure σ and holomorphic cubic differential tq?
• RQ2 Is the solution to the immersion problem unique for small t > 0, or are there multiple distinct minimal Lagrangian immersions for the same (σ, tq)?
• RQ3 What is the asymptotic behavior of the minimal Lagrangian immersion as t → 0, and how does the induced metric e^u g_σ behave in this limit?
• RQ4 Can the surface area of the induced metric be related to a known geometric structure on Teichmüller space, such as the Weil-Petersson metric?
• RQ5 What is the second variation of the area functional along the solution curve γ(t), and how does it relate to the Weil-Petersson norm of the cubic differential q?

Key findings

• For any conformal structure σ and non-zero holomorphic cubic differential q on a closed surface of genus g ≥ 2, there exists a minimal Lagrangian immersion of the universal cover into ℂH² with induced metric e^u g_σ.
• The solution u(t) to the PDE Δu + 2 - 2e^u - 16t²‖q‖²e^{-2u} = 0 exists and is unique near t = 0, forming a smooth curve γ(t) in the solution space.
• As t → 0, the solution u(t) → 0, and the induced metric e^u g_σ → g_σ, the hyperbolic metric of constant curvature -1.
• The surface area functional 𝒜(σ, q) = -∫_Σ e^u dA_σ has vanishing first derivative at t = 0 and second derivative equal to 16∫_Σ ‖q‖² dA_σ.
• The second variation of the area functional is proportional to the Weil-Petersson pairing of the cubic differential q, with 𝒜''(0) = 16⟨q, q⟩_WP.
• The area functional 𝒜 serves as a potential function for the Weil-Petersson metric on the space of holomorphic cubic differentials, with the second variation equal to 16 times the Weil-Petersson norm of q.