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 CP 2 2 Minimal Lagrangian surfaces in CP 2 3 Hamiltonian stationary Lagrangian surfaces in CP 2 4 Lagrangian Bonnet pairs in CP 2 5 Stability

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in CP 2

The Lagrangian surfaces theory in CP 2 has already been very rich. In this talk, we will not cover Good coordinate of Lagrangian surfaces in CP 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 CP 2

Examples of Lagrangian surfaces in CP 2

(CP 2, g, J, ω) the complex projective plane f : Σ → CP 2 Lagrangian surface “Lagrangian ”⇐ ⇒

def f ∗ω = 0 ⇔ Jf∗TΣ⊥f∗TΣ

π : S5 ⊂ C3 → CP 2 Hopf projection

1 Totally geodesic RP 2:

RP 2 = {π(z1, z2, z3) ∈ CP 2|zi = ¯ zi, 1 ≤ i ≤ 3}.

2 Clifford torus:

T 2 = {π(z1, z2, z3) ∈ CP 2||z1|2 = |z2|2 = |z3|2 = 1 3}.

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in CP 2

Examples of Lagrangian surfaces in CP 2

(CP 2, g, J, ω) the complex projective plane f : Σ → CP 2 Lagrangian surface “Lagrangian ”⇐ ⇒

def f ∗ω = 0 ⇔ Jf∗TΣ⊥f∗TΣ

π : S5 ⊂ C3 → CP 2 Hopf projection

1 Totally geodesic RP 2:

RP 2 = {π(z1, z2, z3) ∈ CP 2|zi = ¯ zi, 1 ≤ i ≤ 3}.

2 Clifford torus:

T 2 = {π(z1, z2, z3) ∈ CP 2||z1|2 = |z2|2 = |z3|2 = 1 3}.

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in CP 2

Examples of Lagrangian surfaces in CP 2

(CP 2, g, J, ω) the complex projective plane f : Σ → CP 2 Lagrangian surface “Lagrangian ”⇐ ⇒

def f ∗ω = 0 ⇔ Jf∗TΣ⊥f∗TΣ

π : S5 ⊂ C3 → CP 2 Hopf projection

1 Totally geodesic RP 2:

RP 2 = {π(z1, z2, z3) ∈ CP 2|zi = ¯ zi, 1 ≤ i ≤ 3}.

2 Clifford torus:

T 2 = {π(z1, z2, z3) ∈ CP 2||z1|2 = |z2|2 = |z3|2 = 1 3}.

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in CP 2

Examples of Lagrangian surfaces in CP 2

(CP 2, g, J, ω) the complex projective plane f : Σ → CP 2 Lagrangian surface “Lagrangian ”⇐ ⇒

def f ∗ω = 0 ⇔ Jf∗TΣ⊥f∗TΣ

π : S5 ⊂ C3 → CP 2 Hopf projection

1 Totally geodesic RP 2:

RP 2 = {π(z1, z2, z3) ∈ CP 2|zi = ¯ zi, 1 ≤ i ≤ 3}.

2 Clifford torus:

T 2 = {π(z1, z2, z3) ∈ CP 2||z1|2 = |z2|2 = |z3|2 = 1 3}.

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in CP 2

Lagrangian Surfaces in CP 2

(CP 2, g, J, ω) the complex projective plane f : Σ → CP 2 an oriented Lagrangian surface with the induce metric g = 2eudzd¯ z “Lagrangian ”⇐ ⇒

def f ∗ω = 0 ⇔ Jf∗TΣ⊥f∗TΣ

A.-M. Li and C.P. Wang, Geometry of surfaces in CP 2, preprint, 1999.

• C. Wang, The classification of homogeneous surfaces in CP 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 CP 2

Lagrangian Surfaces in CP 2

W.l.g., we always can choose a local horizontal lift F to S5, i.e., Fz · ¯ F = 0. The metric g is conformal = ⇒ σ = (F, Fz, F¯

z) Hermitian orthogonal

σz = σU, σ¯

z = σV,

U¯

z − Vz = [U, V] ⇐

⇒ u, φ, ψ satisfy φ¯

z + ¯

φz = 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ. Φ := e−uFz¯

z · F¯ zdz := φdz,

Ψ := Fzz · F¯

zdz3 := ψdz3.

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in CP 2

Lagrangian Surfaces in CP 2

W.l.g., we always can choose a local horizontal lift F to S5, i.e., Fz · ¯ F = 0. The metric g is conformal = ⇒ σ = (F, Fz, F¯

z) Hermitian orthogonal

σz = σU, σ¯

z = σV,

U¯

z − Vz = [U, V] ⇐

⇒ u, φ, ψ satisfy φ¯

z + ¯

φz = 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ. Φ := e−uFz¯

z · F¯ zdz := φdz,

Ψ := Fzz · F¯

zdz3 := ψdz3.

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in CP 2

Lagrangian Surfaces in CP 2

W.l.g., we always can choose a local horizontal lift F to S5, i.e., Fz · ¯ F = 0. The metric g is conformal = ⇒ σ = (F, Fz, F¯

z) Hermitian orthogonal

σz = σU, σ¯

z = σV,

U¯

z − Vz = [U, V] ⇐

⇒ u, φ, ψ satisfy φ¯

z + ¯

φz = 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ. Φ := e−uFz¯

z · F¯ zdz := φdz,

Ψ := Fzz · F¯

zdz3 := ψdz3.

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in CP 2

Lagrangian Surfaces in CP 2

W.l.g., we always can choose a local horizontal lift F to S5, i.e., Fz · ¯ F = 0. The metric g is conformal = ⇒ σ = (F, Fz, F¯

z) Hermitian orthogonal

σz = σU, σ¯

z = σV,

U¯

z − Vz = [U, V] ⇐

⇒ u, φ, ψ satisfy φ¯

z + ¯

φz = 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ. Φ := e−uFz¯

z · F¯ zdz := φdz,

Ψ := Fzz · F¯

zdz3 := ψdz3.

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in CP 2

Geometry of Φ and Ψ

Φ := e−uFz¯

z · F¯ zdz := φdz

Ψ := Fzz · F¯

zdz3 := ψdz3

Bonnet theorem A Lagrangian surface in CP 2 is locally determined by {u, φ, ψ} satisfying the following equations φ¯

z + ¯

φz = 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ. 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. αH := ω(H, ·) = i(Φ − ¯ Φ).

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in CP 2

Geometry of Φ and Ψ

Φ := e−uFz¯

z · F¯ zdz := φdz

Ψ := Fzz · F¯

zdz3 := ψdz3

Bonnet theorem A Lagrangian surface in CP 2 is locally determined by {u, φ, ψ} satisfying the following equations φ¯

z + ¯

φz = 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ. 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. αH := ω(H, ·) = i(Φ − ¯ Φ).

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in CP 2

Geometry of Φ and Ψ

Φ := e−uFz¯

z · F¯ zdz := φdz

Ψ := Fzz · F¯

zdz3 := ψdz3

Bonnet theorem A Lagrangian surface in CP 2 is locally determined by {u, φ, ψ} satisfying the following equations φ¯

z + ¯

φz = 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ. 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. αH := ω(H, ·) = i(Φ − ¯ Φ).

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in CP 2

Geometry of Φ and Ψ

Φ := e−uFz¯

z · F¯ zdz := φdz

Ψ := Fzz · F¯

zdz3 := ψdz3

Bonnet theorem A Lagrangian surface in CP 2 is locally determined by {u, φ, ψ} satisfying the following equations φ¯

z + ¯

φz = 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ. 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. αH := ω(H, ·) = i(Φ − ¯ Φ).

On Lagrangian surfaces in the complex projective plane Geometry of surfaces in CP 2

Geometry of Φ and Ψ

Φ := e−uFz¯

z · F¯ zdz := φdz

Ψ := Fzz · F¯

zdz3 := ψdz3

Bonnet theorem A Lagrangian surface in CP 2 is locally determined by {u, φ, ψ} satisfying the following equations φ¯

z + ¯

φz = 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ. 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. αH := ω(H, ·) = i(Φ − ¯ Φ).

On Lagrangian surfaces in the complex projective plane Minimal Lagrangian surfaces in CP 2

Minimal Lagrangian surfaces in CP 2

Minimal Lagrangian ⇐ ⇒ Φ ≡ 0 = ⇒ uz¯

z

= e−2u|ψ|2 − eu, ψ¯

z

= 0, which are invariant under the transformation Ψ → eitΨ for t ∈ R. C ˜ Σ

SL

• C3
• ˜

Σ

• min. Leg.
• S5
• Σ
• min. Lag. CP 2

It gives rise to a local model of singular special Lagrangian 3-folds in Calabi-Yau threefolds.

On Lagrangian surfaces in the complex projective plane Minimal Lagrangian surfaces in CP 2

Minimal Lagrangian surfaces in CP 2

Minimal Lagrangian ⇐ ⇒ Φ ≡ 0 = ⇒ uz¯

z

= e−2u|ψ|2 − eu, ψ¯

z

= 0, which are invariant under the transformation Ψ → eitΨ for t ∈ R. C ˜ Σ

SL

• C3
• ˜

Σ

• min. Leg.
• S5
• Σ
• min. Lag. CP 2

It gives rise to a local model of singular special Lagrangian 3-folds in Calabi-Yau threefolds.

On Lagrangian surfaces in the complex projective plane Minimal Lagrangian surfaces in CP 2

CMC surfaces in R3 and minimal Lagrangian surfaces in CP 2

uz¯

z = e−u|Q|2 − eu,

Q¯

z = 0.

Q = Qdz2 Hopf differential g(Σ) = 0 ⇒ Σ = the round sphere normalize Q = 1: uz¯

z = e−u−eu

(Sinh-Gordon equ.) PDE → ODE (Abresch) CMC tori are constructed in terms of theta functions (Bobenko) CMC surfaces of higher genus by gluing constructions (Kapouleas) uz¯

z = e−2u|ψ|2 − eu,

ψ¯

z = 0.

Ψ = ψdz3 cubic Hopf differential g(Σ) = 0 ⇒ Σ totally geodesic normalize ψ = −1: uz¯

z = e−2u−eu

(Tzitz´ eica equ.) . PDE → ODE (Castro-Urbano) Minimal Lagrangian tori are constructed in terms of Prym theta functions (M.-Ma)

• Min. Lag. surfaces of higher

genus by gluing constructions (Haskins-Kapouleas)

On Lagrangian surfaces in the complex projective plane Minimal Lagrangian surfaces in CP 2

CMC surfaces in R3 and minimal Lagrangian surfaces in CP 2

uz¯

z = e−u|Q|2 − eu,

Q¯

z = 0.

Q = Qdz2 Hopf differential g(Σ) = 0 ⇒ Σ = the round sphere normalize Q = 1: uz¯

z = e−u−eu

(Sinh-Gordon equ.) PDE → ODE (Abresch) CMC tori are constructed in terms of theta functions (Bobenko) CMC surfaces of higher genus by gluing constructions (Kapouleas) uz¯

z = e−2u|ψ|2 − eu,

ψ¯

z = 0.

Ψ = ψdz3 cubic Hopf differential g(Σ) = 0 ⇒ Σ totally geodesic normalize ψ = −1: uz¯

z = e−2u−eu

(Tzitz´ eica equ.) . PDE → ODE (Castro-Urbano) Minimal Lagrangian tori are constructed in terms of Prym theta functions (M.-Ma)

• Min. Lag. surfaces of higher

genus by gluing constructions (Haskins-Kapouleas)

On Lagrangian surfaces in the complex projective plane Minimal Lagrangian surfaces in CP 2

CMC surfaces in R3 and minimal Lagrangian surfaces in CP 2

uz¯

z = e−u|Q|2 − eu,

Q¯

z = 0.

Q = Qdz2 Hopf differential g(Σ) = 0 ⇒ Σ = the round sphere normalize Q = 1: uz¯

z = e−u−eu

(Sinh-Gordon equ.) PDE → ODE (Abresch) CMC tori are constructed in terms of theta functions (Bobenko) CMC surfaces of higher genus by gluing constructions (Kapouleas) uz¯

z = e−2u|ψ|2 − eu,

ψ¯

z = 0.

Ψ = ψdz3 cubic Hopf differential g(Σ) = 0 ⇒ Σ totally geodesic normalize ψ = −1: uz¯

z = e−2u−eu

(Tzitz´ eica equ.) . PDE → ODE (Castro-Urbano) Minimal Lagrangian tori are constructed in terms of Prym theta functions (M.-Ma)

• Min. Lag. surfaces of higher

genus by gluing constructions (Haskins-Kapouleas)

On Lagrangian surfaces in the complex projective plane Minimal Lagrangian surfaces in CP 2

A CMC rotational surface in R3 is locally congruent either a plane, a catenoid, a circular cylinder, a sphere, an unduloid, or a nodoid, which is called a Delaunay surface. Delaunay surfaces play a prominent role in the study of non-compact complete CMC surfaces because they constitute the simplest possible end behavior. A famous result by Korevaar, Kusner and Solomon, building on the deep “preparatory work”of Meeks, asserts that a properly embedded end of a CMC surface is a Delaunay end. Hence a CMC trinoid with embedded ends asymptotically approaches to Delaunay surfaces. Question What plays the role of Delaunay in minimal Lagrangian case? There does not seem to be any “preparatory work”like Meek’s or Korevar-Kusner-Soloman’s.

On Lagrangian surfaces in the complex projective plane Minimal Lagrangian surfaces in CP 2

A CMC rotational surface in R3 is locally congruent either a plane, a catenoid, a circular cylinder, a sphere, an unduloid, or a nodoid, which is called a Delaunay surface. Delaunay surfaces play a prominent role in the study of non-compact complete CMC surfaces because they constitute the simplest possible end behavior. A famous result by Korevaar, Kusner and Solomon, building on the deep “preparatory work”of Meeks, asserts that a properly embedded end of a CMC surface is a Delaunay end. Hence a CMC trinoid with embedded ends asymptotically approaches to Delaunay surfaces. Question What plays the role of Delaunay in minimal Lagrangian case? There does not seem to be any “preparatory work”like Meek’s or Korevar-Kusner-Soloman’s.

On Lagrangian surfaces in the complex projective plane Minimal Lagrangian surfaces in CP 2

A CMC rotational surface in R3 is locally congruent either a plane, a catenoid, a circular cylinder, a sphere, an unduloid, or a nodoid, which is called a Delaunay surface. Delaunay surfaces play a prominent role in the study of non-compact complete CMC surfaces because they constitute the simplest possible end behavior. A famous result by Korevaar, Kusner and Solomon, building on the deep “preparatory work”of Meeks, asserts that a properly embedded end of a CMC surface is a Delaunay end. Hence a CMC trinoid with embedded ends asymptotically approaches to Delaunay surfaces. Question What plays the role of Delaunay in minimal Lagrangian case? There does not seem to be any “preparatory work”like Meek’s or Korevar-Kusner-Soloman’s.

On Lagrangian surfaces in the complex projective plane Minimal Lagrangian surfaces in CP 2

A CMC rotational surface in R3 is locally congruent either a plane, a catenoid, a circular cylinder, a sphere, an unduloid, or a nodoid, which is called a Delaunay surface. Delaunay surfaces play a prominent role in the study of non-compact complete CMC surfaces because they constitute the simplest possible end behavior. A famous result by Korevaar, Kusner and Solomon, building on the deep “preparatory work”of Meeks, asserts that a properly embedded end of a CMC surface is a Delaunay end. Hence a CMC trinoid with embedded ends asymptotically approaches to Delaunay surfaces. Question What plays the role of Delaunay in minimal Lagrangian case? There does not seem to be any “preparatory work”like Meek’s or Korevar-Kusner-Soloman’s.

On Lagrangian surfaces in the complex projective plane Minimal Lagrangian surfaces in CP 2

Further questions on minimal Lagrangian surfaces in CP 2

CMC trinoids with embedded ends are constructed by DPW method by Dorfmeister-Wu and Schmitt-Kilian-Kobayashi-Rossman. Question. How to construct minimal Lagrangian trinoids in CP 2? Is it possible to construct them by the DPW method?

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

Hamiltonian minimal ( Hamiltonian stationary Lagrangian) submanifolds

Lawson-Simons: Any stable minimal submanifolds in CP n is a complex submanifold. = ⇒ The minimal Lagrangian submanifolds in CP n cannot be stable under general variations. (Y.G. Oh) Hamiltonian minimal (or called “Hamiltonian stationary Lagrangian”by Schoen-Wolfson) submanifolds of a K¨ ahler manifold are the critical points of the volume functional on Lagrangian submanifolds under any (compactly supported) Hamiltonian variations. It is a generalization for minimal Lagrangian submanifolds. A Lagrangian immersion f is H-minimal/HSL ⇐ ⇒ δαH = 0.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

Hamiltonian minimal ( Hamiltonian stationary Lagrangian) submanifolds

Lawson-Simons: Any stable minimal submanifolds in CP n is a complex submanifold. = ⇒ The minimal Lagrangian submanifolds in CP n cannot be stable under general variations. (Y.G. Oh) Hamiltonian minimal (or called “Hamiltonian stationary Lagrangian”by Schoen-Wolfson) submanifolds of a K¨ ahler manifold are the critical points of the volume functional on Lagrangian submanifolds under any (compactly supported) Hamiltonian variations. It is a generalization for minimal Lagrangian submanifolds. A Lagrangian immersion f is H-minimal/HSL ⇐ ⇒ δαH = 0.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

Hamiltonian minimal ( Hamiltonian stationary Lagrangian) submanifolds

Lawson-Simons: Any stable minimal submanifolds in CP n is a complex submanifold. = ⇒ The minimal Lagrangian submanifolds in CP n cannot be stable under general variations. (Y.G. Oh) Hamiltonian minimal (or called “Hamiltonian stationary Lagrangian”by Schoen-Wolfson) submanifolds of a K¨ ahler manifold are the critical points of the volume functional on Lagrangian submanifolds under any (compactly supported) Hamiltonian variations. It is a generalization for minimal Lagrangian submanifolds. A Lagrangian immersion f is H-minimal/HSL ⇐ ⇒ δαH = 0.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

Hamiltonian minimal ( Hamiltonian stationary Lagrangian) submanifolds

Lawson-Simons: Any stable minimal submanifolds in CP n is a complex submanifold. = ⇒ The minimal Lagrangian submanifolds in CP n cannot be stable under general variations. (Y.G. Oh) Hamiltonian minimal (or called “Hamiltonian stationary Lagrangian”by Schoen-Wolfson) submanifolds of a K¨ ahler manifold are the critical points of the volume functional on Lagrangian submanifolds under any (compactly supported) Hamiltonian variations. It is a generalization for minimal Lagrangian submanifolds. A Lagrangian immersion f is H-minimal/HSL ⇐ ⇒ δαH = 0.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

Hamiltonian minimal ( Hamiltonian stationary Lagrangian) submanifolds

Lawson-Simons: Any stable minimal submanifolds in CP n is a complex submanifold. = ⇒ The minimal Lagrangian submanifolds in CP n cannot be stable under general variations. (Y.G. Oh) Hamiltonian minimal (or called “Hamiltonian stationary Lagrangian”by Schoen-Wolfson) submanifolds of a K¨ ahler manifold are the critical points of the volume functional on Lagrangian submanifolds under any (compactly supported) Hamiltonian variations. It is a generalization for minimal Lagrangian submanifolds. A Lagrangian immersion f is H-minimal/HSL ⇐ ⇒ δαH = 0.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

Hamiltonian minimal ( Hamiltonian stationary Lagrangian) submanifolds

Lawson-Simons: Any stable minimal submanifolds in CP n is a complex submanifold. = ⇒ The minimal Lagrangian submanifolds in CP n cannot be stable under general variations. (Y.G. Oh) Hamiltonian minimal (or called “Hamiltonian stationary Lagrangian”by Schoen-Wolfson) submanifolds of a K¨ ahler manifold are the critical points of the volume functional on Lagrangian submanifolds under any (compactly supported) Hamiltonian variations. It is a generalization for minimal Lagrangian submanifolds. A Lagrangian immersion f is H-minimal/HSL ⇐ ⇒ δαH = 0.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

HSL Surfaces in CP 2

A Lagrangian surface in CP 2 is Hamiltonian stationary if and only if Φ is a holomorphic. ⇒ The compatibility equations for HSL surfaces in CP 2 are φ¯

z

= 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ, Remark 1.The above equations are invariant w.r.t. the transformation Φ → νΦ, Ψ → νΨ for ν ∈ S1.

• 2. αH = i(Φ − ¯

Φ).

• 3. Any HSL sphere in CP 2 is totally geodesic.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

HSL Surfaces in CP 2

A Lagrangian surface in CP 2 is Hamiltonian stationary if and only if Φ is a holomorphic. ⇒ The compatibility equations for HSL surfaces in CP 2 are φ¯

z

= 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ, Remark 1.The above equations are invariant w.r.t. the transformation Φ → νΦ, Ψ → νΨ for ν ∈ S1.

• 2. αH = i(Φ − ¯

Φ).

• 3. Any HSL sphere in CP 2 is totally geodesic.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

HSL Surfaces in CP 2

A Lagrangian surface in CP 2 is Hamiltonian stationary if and only if Φ is a holomorphic. ⇒ The compatibility equations for HSL surfaces in CP 2 are φ¯

z

= 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ, Remark 1.The above equations are invariant w.r.t. the transformation Φ → νΦ, Ψ → νΨ for ν ∈ S1.

• 2. αH = i(Φ − ¯

Φ).

• 3. Any HSL sphere in CP 2 is totally geodesic.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

HSL Surfaces in CP 2

A Lagrangian surface in CP 2 is Hamiltonian stationary if and only if Φ is a holomorphic. ⇒ The compatibility equations for HSL surfaces in CP 2 are φ¯

z

= 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ, Remark 1.The above equations are invariant w.r.t. the transformation Φ → νΦ, Ψ → νΨ for ν ∈ S1.

• 2. αH = i(Φ − ¯

Φ).

• 3. Any HSL sphere in CP 2 is totally geodesic.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

HSL Surfaces in CP 2

A Lagrangian surface in CP 2 is Hamiltonian stationary if and only if Φ is a holomorphic. ⇒ The compatibility equations for HSL surfaces in CP 2 are φ¯

z

= 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ, Remark 1.The above equations are invariant w.r.t. the transformation Φ → νΦ, Ψ → νΨ for ν ∈ S1.

• 2. αH = i(Φ − ¯

Φ).

• 3. Any HSL sphere in CP 2 is totally geodesic.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

Hamiltonian stationary Lagrangian surfaces in CP 2

H´ elein-Romon formulated HSL surfaces in 2-dim. Hermitian symmetric spaces are solutions of integrable systems as the vanishing of the curvature of a certain 1-parameter family of flat connections. Terng explained the HSL surfaces in CP 2 as the second elliptic SU(3)/SU(2)-system. M., H´ elein-Romon: Any HSL torus in CP 2 can be constructed from a pair of commuting Hamiltonian ODEs on a finite dimensional subspaces of a certain loop Lie algebra, i.e., of finite type. M.-Schemies, Mironov: reduced PDE to ODE and gave new explicit examples for HSL for tori which are invariant under a one-parameter group of isometries of CP 2. Hunter-McIntosh studied the spectral data of HSL tori in CP 2.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

Hamiltonian stationary Lagrangian surfaces in CP 2

H´ elein-Romon formulated HSL surfaces in 2-dim. Hermitian symmetric spaces are solutions of integrable systems as the vanishing of the curvature of a certain 1-parameter family of flat connections. Terng explained the HSL surfaces in CP 2 as the second elliptic SU(3)/SU(2)-system. M., H´ elein-Romon: Any HSL torus in CP 2 can be constructed from a pair of commuting Hamiltonian ODEs on a finite dimensional subspaces of a certain loop Lie algebra, i.e., of finite type. M.-Schemies, Mironov: reduced PDE to ODE and gave new explicit examples for HSL for tori which are invariant under a one-parameter group of isometries of CP 2. Hunter-McIntosh studied the spectral data of HSL tori in CP 2.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

Hamiltonian stationary Lagrangian surfaces in CP 2

H´ elein-Romon formulated HSL surfaces in 2-dim. Hermitian symmetric spaces are solutions of integrable systems as the vanishing of the curvature of a certain 1-parameter family of flat connections. Terng explained the HSL surfaces in CP 2 as the second elliptic SU(3)/SU(2)-system. M., H´ elein-Romon: Any HSL torus in CP 2 can be constructed from a pair of commuting Hamiltonian ODEs on a finite dimensional subspaces of a certain loop Lie algebra, i.e., of finite type. M.-Schemies, Mironov: reduced PDE to ODE and gave new explicit examples for HSL for tori which are invariant under a one-parameter group of isometries of CP 2. Hunter-McIntosh studied the spectral data of HSL tori in CP 2.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

Hamiltonian stationary Lagrangian surfaces in CP 2

H´ elein-Romon formulated HSL surfaces in 2-dim. Hermitian symmetric spaces are solutions of integrable systems as the vanishing of the curvature of a certain 1-parameter family of flat connections. Terng explained the HSL surfaces in CP 2 as the second elliptic SU(3)/SU(2)-system. M., H´ elein-Romon: Any HSL torus in CP 2 can be constructed from a pair of commuting Hamiltonian ODEs on a finite dimensional subspaces of a certain loop Lie algebra, i.e., of finite type. M.-Schemies, Mironov: reduced PDE to ODE and gave new explicit examples for HSL for tori which are invariant under a one-parameter group of isometries of CP 2. Hunter-McIntosh studied the spectral data of HSL tori in CP 2.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

Hamiltonian stationary Lagrangian surfaces in CP 2

H´ elein-Romon formulated HSL surfaces in 2-dim. Hermitian symmetric spaces are solutions of integrable systems as the vanishing of the curvature of a certain 1-parameter family of flat connections. Terng explained the HSL surfaces in CP 2 as the second elliptic SU(3)/SU(2)-system. M., H´ elein-Romon: Any HSL torus in CP 2 can be constructed from a pair of commuting Hamiltonian ODEs on a finite dimensional subspaces of a certain loop Lie algebra, i.e., of finite type. M.-Schemies, Mironov: reduced PDE to ODE and gave new explicit examples for HSL for tori which are invariant under a one-parameter group of isometries of CP 2. Hunter-McIntosh studied the spectral data of HSL tori in CP 2.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

Hamiltonian stationary Lagrangian surfaces in CP 2

H´ elein-Romon formulated HSL surfaces in 2-dim. Hermitian symmetric spaces are solutions of integrable systems as the vanishing of the curvature of a certain 1-parameter family of flat connections. Terng explained the HSL surfaces in CP 2 as the second elliptic SU(3)/SU(2)-system. M., H´ elein-Romon: Any HSL torus in CP 2 can be constructed from a pair of commuting Hamiltonian ODEs on a finite dimensional subspaces of a certain loop Lie algebra, i.e., of finite type. M.-Schemies, Mironov: reduced PDE to ODE and gave new explicit examples for HSL for tori which are invariant under a one-parameter group of isometries of CP 2. Hunter-McIntosh studied the spectral data of HSL tori in CP 2.

On Lagrangian surfaces in the complex projective plane Hamiltonian stationary Lagrangian surfaces in CP 2

S1-invariant HSL tori in CP 2

On a nonminimal HSL torus, φ¯

z = 0 and φ is a bounded elliptic function,

thus a nonzero constant. By a conformal change of coordinate one can always normalize φ = 1. Then the equations become ψ¯

z

= −(eu)z, uz¯

z + eu + 1 − e−2u|ψ|2

= 0. Now we assume that u and ψ depend only on one variable x. Then ψ = −eu + C, where C is a nonzero constant. Thus we have an ODE 1 4u′′ + eu + (C + ¯ C)e−u − |C|2e−2u = 0. Moreover the lift F(−, y) satisfies the following linear differential vector-valued equation Fyyy − 4iFyy + 2aFy − i(C + ¯ C)F = 0.

On Lagrangian surfaces in the complex projective plane Lagrangian Bonnet pairs in CP 2

Twistor harmonic surfaces

A surface in CP 2 is called twistor harmonic if its twistor lift to SU(3)/T 2 is harmonic map. Proposition(Castro-Urbano) A Lagrangian immersion of an oriented surface is twistor harmonic if and

• nly if Ψ is holomorphic.

Castro-Urbano classified all compact twistor harmonic non-minimal Lagrangian surfaces in CP 2.

On Lagrangian surfaces in the complex projective plane Lagrangian Bonnet pairs in CP 2

Classical Bonnet problem

Classical Bonnet Problem Whether the metric eu and the mean curvature function H are suffice to determine a surface in R3 up to rigidity motions? Around a non-umbilical point, a surface in R3 is determined uniquely by the metric eu and the mean curvature function H, except three classes of surfaces: CMC surfaces, Bonnet surfaces and Bonnet pairs. Much less is known about Bonnet pairs. Lawson-Tribuzy showed that for compact oriented surfaces in R3 with nonconstant mean curvature, there are at most two surfaces with the given metric and mean curvature. There are no Bonnet pairs of genus zero in R3. It is open whether compact Bonnet pairs exist.

On Lagrangian surfaces in the complex projective plane Lagrangian Bonnet pairs in CP 2

Classical Bonnet problem

Classical Bonnet Problem Whether the metric eu and the mean curvature function H are suffice to determine a surface in R3 up to rigidity motions? Around a non-umbilical point, a surface in R3 is determined uniquely by the metric eu and the mean curvature function H, except three classes of surfaces: CMC surfaces, Bonnet surfaces and Bonnet pairs. Much less is known about Bonnet pairs. Lawson-Tribuzy showed that for compact oriented surfaces in R3 with nonconstant mean curvature, there are at most two surfaces with the given metric and mean curvature. There are no Bonnet pairs of genus zero in R3. It is open whether compact Bonnet pairs exist.

On Lagrangian surfaces in the complex projective plane Lagrangian Bonnet pairs in CP 2

Classical Bonnet problem

Classical Bonnet Problem Whether the metric eu and the mean curvature function H are suffice to determine a surface in R3 up to rigidity motions? Around a non-umbilical point, a surface in R3 is determined uniquely by the metric eu and the mean curvature function H, except three classes of surfaces: CMC surfaces, Bonnet surfaces and Bonnet pairs. Much less is known about Bonnet pairs. Lawson-Tribuzy showed that for compact oriented surfaces in R3 with nonconstant mean curvature, there are at most two surfaces with the given metric and mean curvature. There are no Bonnet pairs of genus zero in R3. It is open whether compact Bonnet pairs exist.

On Lagrangian surfaces in the complex projective plane Lagrangian Bonnet pairs in CP 2

Classical Bonnet problem

Classical Bonnet Problem Whether the metric eu and the mean curvature function H are suffice to determine a surface in R3 up to rigidity motions? Around a non-umbilical point, a surface in R3 is determined uniquely by the metric eu and the mean curvature function H, except three classes of surfaces: CMC surfaces, Bonnet surfaces and Bonnet pairs. Much less is known about Bonnet pairs. Lawson-Tribuzy showed that for compact oriented surfaces in R3 with nonconstant mean curvature, there are at most two surfaces with the given metric and mean curvature. There are no Bonnet pairs of genus zero in R3. It is open whether compact Bonnet pairs exist.

On Lagrangian surfaces in the complex projective plane Lagrangian Bonnet pairs in CP 2

Classical Bonnet problem

Classical Bonnet Problem Whether the metric eu and the mean curvature function H are suffice to determine a surface in R3 up to rigidity motions? Around a non-umbilical point, a surface in R3 is determined uniquely by the metric eu and the mean curvature function H, except three classes of surfaces: CMC surfaces, Bonnet surfaces and Bonnet pairs. Much less is known about Bonnet pairs. Lawson-Tribuzy showed that for compact oriented surfaces in R3 with nonconstant mean curvature, there are at most two surfaces with the given metric and mean curvature. There are no Bonnet pairs of genus zero in R3. It is open whether compact Bonnet pairs exist.

On Lagrangian surfaces in the complex projective plane Lagrangian Bonnet pairs in CP 2

Classical Bonnet problem

Classical Bonnet Problem Whether the metric eu and the mean curvature function H are suffice to determine a surface in R3 up to rigidity motions? Around a non-umbilical point, a surface in R3 is determined uniquely by the metric eu and the mean curvature function H, except three classes of surfaces: CMC surfaces, Bonnet surfaces and Bonnet pairs. Much less is known about Bonnet pairs. Lawson-Tribuzy showed that for compact oriented surfaces in R3 with nonconstant mean curvature, there are at most two surfaces with the given metric and mean curvature. There are no Bonnet pairs of genus zero in R3. It is open whether compact Bonnet pairs exist.

On Lagrangian surfaces in the complex projective plane Lagrangian Bonnet pairs in CP 2

Lagrangian Bonnet pairs

Lagrangian Bonnet Problem Whether the metric eu and the mean curvature form Φ are suffice to determine a Lagrangian surface in CP 2 up to rigidity motions? Lagrangian Bonnet pairs in CP 2 are two non-congruent isometric surfaces with the same mean curvature 1-form Φ. Theorem 4.1 (He-M.). Let Σ be a compact oriented Lagrangian surface in CP 2. If Σ is not twistor harmonic, then there exists at most two noncongruent isometric immersions

• f Σ in CP 2 with the mean curvature form Φ.

This theorem also holds in C2 and CH2.

On Lagrangian surfaces in the complex projective plane Lagrangian Bonnet pairs in CP 2

Lagrangian Bonnet pairs

Lagrangian Bonnet Problem Whether the metric eu and the mean curvature form Φ are suffice to determine a Lagrangian surface in CP 2 up to rigidity motions? Lagrangian Bonnet pairs in CP 2 are two non-congruent isometric surfaces with the same mean curvature 1-form Φ. Theorem 4.1 (He-M.). Let Σ be a compact oriented Lagrangian surface in CP 2. If Σ is not twistor harmonic, then there exists at most two noncongruent isometric immersions

• f Σ in CP 2 with the mean curvature form Φ.

This theorem also holds in C2 and CH2.

On Lagrangian surfaces in the complex projective plane Lagrangian Bonnet pairs in CP 2

Sketch of the proof

φ¯

z + ¯

φz = 0, uz¯

z + eu + |φ|2 − e−2u|ψ|2

= 0, e−uψ¯

z

= φz − uzφ. Suppose that there are 3 isometric noncongruent Lagrangian immersions fk : Σ → CP 2 (k = 1, 2, 3) with coinciding mean curvature one-form Φ. Ψij = Ψi − Ψj (1 ≤ i, j ≤ 3) is holomorphic on Σ. Moreover, |Ψi| = |Ψj| ⇒ the zero of Ψk coincide. Denote U = {p ∈ Σ|Ψk(p) = 0}. If the three immersions fk(k = 1, 2, 3) are mutually noncongruent, then it follows from the fundamental theorem of algebra that ∂2 ∂z∂¯ z log ψk = |∂ψk ∂¯ z |2.

On Lagrangian surfaces in the complex projective plane Lagrangian Bonnet pairs in CP 2

Let ψ2 = ψ1eiθ. Then Q := ψ1−ψ2

ψ1

= 1 − eiθ is well defined on Σ\U. Notice △ log Q = △ log |Q| + i△ arg Q ≤ 0 on Σ\U, ⇒ arg Q can be extended to a (bounded) harmonic function on Σ. Then Q is a constant hence Ψ1 is holomorphic which contradicts with f1 is not twistor harmonic. Remark. Certain S1-invariant HSL tori in CP 2 provide examples of compact Lagrangian Bonnet pairs in CP 2.

On Lagrangian surfaces in the complex projective plane Lagrangian Bonnet pairs in CP 2

Further question on Lagrangian Bonnet pairs

The classical theory of Bonnet pairs in R3 is related to the theory of isothermic surfaces and belongs also the geometry described by integrable systems. Question. Does there exist a Lagrangian version of isothemic surfaces in the complex space forms? Do Lagrangian Bonnet pairs relate to integrable systems?

On Lagrangian surfaces in the complex projective plane Stability

Minimal surfaces in S3 and minimal Lagrangian surfaces in CP 2

• J. Simons: Any compact minimal surface Σ in S3(1) is unstable and

Ind(Σ) ≥ 1. “=”holds ⇐ ⇒ Σ is totally geodesic. Urbano: Let Σ be a compact orientable nontotally geodesic minimal surface in S3(1). Then Ind(Σ) ≥ 5. “= ”holds ⇐ ⇒ Σ is the Clifford torus. Lawson-Simons: Any minimal Lagrangian surface in CP 2 is unstable. Urbano: For any minimal Lagrangian compact orientable surface in CP 2, Ind(Σ) ≥ 2. “= ”holds ⇐ ⇒Σ is the Clifford torus. Urbano: For any minimal Lagrangian compact nonorientable surface of CP 2, Ind(Σ) ≥ 3. “= ”holds ⇐ ⇒ Σ is the totally geodesic RP 2.

On Lagrangian surfaces in the complex projective plane Stability

Minimal surfaces in S3 and minimal Lagrangian surfaces in CP 2

• J. Simons: Any compact minimal surface Σ in S3(1) is unstable and

Ind(Σ) ≥ 1. “=”holds ⇐ ⇒ Σ is totally geodesic. Urbano: Let Σ be a compact orientable nontotally geodesic minimal surface in S3(1). Then Ind(Σ) ≥ 5. “= ”holds ⇐ ⇒ Σ is the Clifford torus. Lawson-Simons: Any minimal Lagrangian surface in CP 2 is unstable. Urbano: For any minimal Lagrangian compact orientable surface in CP 2, Ind(Σ) ≥ 2. “= ”holds ⇐ ⇒Σ is the Clifford torus. Urbano: For any minimal Lagrangian compact nonorientable surface of CP 2, Ind(Σ) ≥ 3. “= ”holds ⇐ ⇒ Σ is the totally geodesic RP 2.

On Lagrangian surfaces in the complex projective plane Stability

Lawson conjecture, proved by S. Brendle The only compact embedded minimal torus in S3(1) is the Clifford torus. Lagrangian version of the Lawson conjecture The only compact embedded minimal Lagrangian torus in CP 2 is the Clifford torus.

On Lagrangian surfaces in the complex projective plane Stability

Lawson conjecture, proved by S. Brendle The only compact embedded minimal torus in S3(1) is the Clifford torus. Lagrangian version of the Lawson conjecture The only compact embedded minimal Lagrangian torus in CP 2 is the Clifford torus.

On Lagrangian surfaces in the complex projective plane

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