On Lagrangian surfaces in the complex projective plane Hui Ma - PowerPoint PPT Presentation

On Lagrangian surfaces in the complex projective plane On Lagrangian surfaces in the complex projective plane Hui Ma Department of Mathematical Sciences Tsinghua University, Beijing, 100084, China hma@math.tsinghua.edu.cn PADGE2012, August 27-30, Leuven

On Lagrangian surfaces in the complex projective plane Contents 1 Geometry of surfaces in C P 2 2 Minimal Lagrangian surfaces in C P 2 3 Hamiltonian stationary Lagrangian surfaces in C P 2 4 Lagrangian Bonnet pairs in C P 2 5 Stability

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 The Lagrangian surfaces theory in C P 2 has already been very rich. In this talk, we will not cover Good coordinate of Lagrangian surfaces in C P 2 Pinching results Simons formula Relation with affine spheres Relation with Painlev´ e III equation ......

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 Examples of Lagrangian surfaces in C P 2 ( C P 2 , g, J, ω ) the complex projective plane f : Σ → C P 2 Lagrangian surface def f ∗ ω = 0 ⇔ Jf ∗ T Σ ⊥ f ∗ T Σ “Lagrangian ” ⇐ ⇒ π : S 5 ⊂ C 3 → C P 2 Hopf projection 1 Totally geodesic R P 2 : R P 2 = { π ( z 1 , z 2 , z 3 ) ∈ C P 2 | z i = ¯ z i , 1 ≤ i ≤ 3 } . 2 Clifford torus: T 2 = { π ( z 1 , z 2 , z 3 ) ∈ C P 2 || z 1 | 2 = | z 2 | 2 = | z 3 | 2 = 1 3 } .

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 Examples of Lagrangian surfaces in C P 2 ( C P 2 , g, J, ω ) the complex projective plane f : Σ → C P 2 Lagrangian surface def f ∗ ω = 0 ⇔ Jf ∗ T Σ ⊥ f ∗ T Σ “Lagrangian ” ⇐ ⇒ π : S 5 ⊂ C 3 → C P 2 Hopf projection 1 Totally geodesic R P 2 : R P 2 = { π ( z 1 , z 2 , z 3 ) ∈ C P 2 | z i = ¯ z i , 1 ≤ i ≤ 3 } . 2 Clifford torus: T 2 = { π ( z 1 , z 2 , z 3 ) ∈ C P 2 || z 1 | 2 = | z 2 | 2 = | z 3 | 2 = 1 3 } .

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 Examples of Lagrangian surfaces in C P 2 ( C P 2 , g, J, ω ) the complex projective plane f : Σ → C P 2 Lagrangian surface def f ∗ ω = 0 ⇔ Jf ∗ T Σ ⊥ f ∗ T Σ “Lagrangian ” ⇐ ⇒ π : S 5 ⊂ C 3 → C P 2 Hopf projection 1 Totally geodesic R P 2 : R P 2 = { π ( z 1 , z 2 , z 3 ) ∈ C P 2 | z i = ¯ z i , 1 ≤ i ≤ 3 } . 2 Clifford torus: T 2 = { π ( z 1 , z 2 , z 3 ) ∈ C P 2 || z 1 | 2 = | z 2 | 2 = | z 3 | 2 = 1 3 } .

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 Examples of Lagrangian surfaces in C P 2 ( C P 2 , g, J, ω ) the complex projective plane f : Σ → C P 2 Lagrangian surface def f ∗ ω = 0 ⇔ Jf ∗ T Σ ⊥ f ∗ T Σ “Lagrangian ” ⇐ ⇒ π : S 5 ⊂ C 3 → C P 2 Hopf projection 1 Totally geodesic R P 2 : R P 2 = { π ( z 1 , z 2 , z 3 ) ∈ C P 2 | z i = ¯ z i , 1 ≤ i ≤ 3 } . 2 Clifford torus: T 2 = { π ( z 1 , z 2 , z 3 ) ∈ C P 2 || z 1 | 2 = | z 2 | 2 = | z 3 | 2 = 1 3 } .

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 Lagrangian Surfaces in C P 2 ( C P 2 , g, J, ω ) the complex projective plane f : Σ → C P 2 an oriented Lagrangian surface with the induce metric g = 2 e u dzd ¯ z def f ∗ ω = 0 ⇔ Jf ∗ T Σ ⊥ f ∗ T Σ “Lagrangian ” ⇐ ⇒ A.-M. Li and C.P. Wang, Geometry of surfaces in C P 2 , preprint, 1999. C. Wang, The classification of homogeneous surfaces in C P 2 , Geometry and topology of submanifolds, X (Beijing/Berlin, 1999), 303-314, 2000.

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 Lagrangian Surfaces in C P 2 W.l.g., we always can choose a local horizontal lift F to S 5 , i.e., F z · ¯ F = 0. The metric g is conformal = ⇒ σ = ( F, F z , F ¯ z ) Hermitian orthogonal σ z = σ U , σ ¯ z = σ V , U ¯ z − V z = [ U , V ] ⇐ ⇒ u, φ, ψ satisfy z + ¯ φ ¯ φ z = 0 , z + e u + | φ | 2 − e − 2 u | ψ | 2 u z ¯ = 0 , e − u ψ ¯ = φ z − u z φ. z z dz 3 := ψdz 3 . Φ := e − u F z ¯ z · F ¯ z dz := φdz, Ψ := F zz · F ¯

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 Lagrangian Surfaces in C P 2 W.l.g., we always can choose a local horizontal lift F to S 5 , i.e., F z · ¯ F = 0. The metric g is conformal = ⇒ σ = ( F, F z , F ¯ z ) Hermitian orthogonal σ z = σ U , σ ¯ z = σ V , U ¯ z − V z = [ U , V ] ⇐ ⇒ u, φ, ψ satisfy z + ¯ φ ¯ φ z = 0 , z + e u + | φ | 2 − e − 2 u | ψ | 2 u z ¯ = 0 , e − u ψ ¯ = φ z − u z φ. z z dz 3 := ψdz 3 . Φ := e − u F z ¯ z · F ¯ z dz := φdz, Ψ := F zz · F ¯

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 Lagrangian Surfaces in C P 2 W.l.g., we always can choose a local horizontal lift F to S 5 , i.e., F z · ¯ F = 0. The metric g is conformal = ⇒ σ = ( F, F z , F ¯ z ) Hermitian orthogonal σ z = σ U , σ ¯ z = σ V , U ¯ z − V z = [ U , V ] ⇐ ⇒ u, φ, ψ satisfy z + ¯ φ ¯ φ z = 0 , z + e u + | φ | 2 − e − 2 u | ψ | 2 u z ¯ = 0 , e − u ψ ¯ = φ z − u z φ. z z dz 3 := ψdz 3 . Φ := e − u F z ¯ z · F ¯ z dz := φdz, Ψ := F zz · F ¯

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 Lagrangian Surfaces in C P 2 W.l.g., we always can choose a local horizontal lift F to S 5 , i.e., F z · ¯ F = 0. The metric g is conformal = ⇒ σ = ( F, F z , F ¯ z ) Hermitian orthogonal σ z = σ U , σ ¯ z = σ V , U ¯ z − V z = [ U , V ] ⇐ ⇒ u, φ, ψ satisfy z + ¯ φ ¯ φ z = 0 , z + e u + | φ | 2 − e − 2 u | ψ | 2 u z ¯ = 0 , e − u ψ ¯ = φ z − u z φ. z z dz 3 := ψdz 3 . Φ := e − u F z ¯ z · F ¯ z dz := φdz, Ψ := F zz · F ¯

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 Geometry of Φ and Ψ z dz 3 := ψdz 3 Φ := e − u F z ¯ z · F ¯ z dz := φdz Ψ := F zz · F ¯ Bonnet theorem A Lagrangian surface in C P 2 is locally determined by { u, φ, ψ } satisfying the following equations z + ¯ φ ¯ φ z = 0 , z + e u + | φ | 2 − e − 2 u | ψ | 2 u z ¯ = 0 , e − u ψ ¯ = φ z − u z φ. z f is minimal ⇐ ⇒ Φ ≡ 0 = ⇒ Ψ is holomorphic. f is Hamiltonian stationary Lagrangian (Hamiltonian minimal) ⇐ ⇒ Φ is holomorphic. f is twistor harmonic ⇐ ⇒ Ψ is holomorphic. Remark. α H := ω ( H, · ) = i (Φ − ¯ Φ) .

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 Geometry of Φ and Ψ z dz 3 := ψdz 3 Φ := e − u F z ¯ z · F ¯ z dz := φdz Ψ := F zz · F ¯ Bonnet theorem A Lagrangian surface in C P 2 is locally determined by { u, φ, ψ } satisfying the following equations z + ¯ φ ¯ φ z = 0 , z + e u + | φ | 2 − e − 2 u | ψ | 2 u z ¯ = 0 , e − u ψ ¯ = φ z − u z φ. z f is minimal ⇐ ⇒ Φ ≡ 0 = ⇒ Ψ is holomorphic. f is Hamiltonian stationary Lagrangian (Hamiltonian minimal) ⇐ ⇒ Φ is holomorphic. f is twistor harmonic ⇐ ⇒ Ψ is holomorphic. Remark. α H := ω ( H, · ) = i (Φ − ¯ Φ) .

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 Geometry of Φ and Ψ z dz 3 := ψdz 3 Φ := e − u F z ¯ z · F ¯ z dz := φdz Ψ := F zz · F ¯ Bonnet theorem A Lagrangian surface in C P 2 is locally determined by { u, φ, ψ } satisfying the following equations z + ¯ φ ¯ φ z = 0 , z + e u + | φ | 2 − e − 2 u | ψ | 2 u z ¯ = 0 , e − u ψ ¯ = φ z − u z φ. z f is minimal ⇐ ⇒ Φ ≡ 0 = ⇒ Ψ is holomorphic. f is Hamiltonian stationary Lagrangian (Hamiltonian minimal) ⇐ ⇒ Φ is holomorphic. f is twistor harmonic ⇐ ⇒ Ψ is holomorphic. Remark. α H := ω ( H, · ) = i (Φ − ¯ Φ) .

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 Geometry of Φ and Ψ z dz 3 := ψdz 3 Φ := e − u F z ¯ z · F ¯ z dz := φdz Ψ := F zz · F ¯ Bonnet theorem A Lagrangian surface in C P 2 is locally determined by { u, φ, ψ } satisfying the following equations z + ¯ φ ¯ φ z = 0 , z + e u + | φ | 2 − e − 2 u | ψ | 2 u z ¯ = 0 , e − u ψ ¯ = φ z − u z φ. z f is minimal ⇐ ⇒ Φ ≡ 0 = ⇒ Ψ is holomorphic. f is Hamiltonian stationary Lagrangian (Hamiltonian minimal) ⇐ ⇒ Φ is holomorphic. f is twistor harmonic ⇐ ⇒ Ψ is holomorphic. Remark. α H := ω ( H, · ) = i (Φ − ¯ Φ) .

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in C P 2 Geometry of Φ and Ψ z dz 3 := ψdz 3 Φ := e − u F z ¯ z · F ¯ z dz := φdz Ψ := F zz · F ¯ Bonnet theorem A Lagrangian surface in C P 2 is locally determined by { u, φ, ψ } satisfying the following equations z + ¯ φ ¯ φ z = 0 , z + e u + | φ | 2 − e − 2 u | ψ | 2 u z ¯ = 0 , e − u ψ ¯ = φ z − u z φ. z f is minimal ⇐ ⇒ Φ ≡ 0 = ⇒ Ψ is holomorphic. f is Hamiltonian stationary Lagrangian (Hamiltonian minimal) ⇐ ⇒ Φ is holomorphic. f is twistor harmonic ⇐ ⇒ Ψ is holomorphic. Remark. α H := ω ( H, · ) = i (Φ − ¯ Φ) .

� � � � � On Lagrangian surfaces in the complex projective plane Minimal Lagrangian surfaces in C P 2 Minimal Lagrangian surfaces in C P 2 Minimal Lagrangian ⇐ ⇒ Φ ≡ 0 = ⇒ e − 2 u | ψ | 2 − e u , u z ¯ = z ψ ¯ = 0 , z which are invariant under the transformation Ψ → e it Ψ for t ∈ R . SL C ˜ C 3 Σ min. Leg. � ˜ S 5 Σ min. Lag. � C P 2 Σ It gives rise to a local model of singular special Lagrangian 3-folds in Calabi-Yau threefolds.