Lagrangian mean curvature flow Ildefonso Castro (joint work with - - PowerPoint PPT Presentation

Lagrangian mean curvature flow

Ildefonso Castro (joint work with Ana M. Lerma)

Departamento de Matem´ aticas Universidad de Ja´ en 23071 Ja´ en, Spain

2012 PADGE Conference on Pure and Applied Differential Geometry Leuven, August 2012

Index

1 Preliminaries

Mean curvature flow (MCF) Lagrangian submanifolds in complex Euclidean space Lagrangian MCF

2 Self-similar solutions for Lagrangian mean curvature flow

Examples Classification of Hamiltonian stationary Lagrangians (HSL)

3 The Clifford torus as a self-shrinker 4 Translating solitons for Lagrangian mean curvature flow

PRELIMINARIES

Mean curvature flow (MCF)

Mean curvature flow (MCF)

F0 : Mn → Rm smooth immersion, M compact

Mean curvature flow (MCF)

F0 : Mn → Rm smooth immersion, M compact MCF with initial condition F0 F : M × [0, T) → Rm, Ft = F(·, t) ∂ ∂t F(p, t) = H(p, t), p ∈ M, t ≥ 0; F(·, 0) = F0 (MCF)

Mean curvature flow (MCF)

F0 : Mn → Rm smooth immersion, M compact MCF with initial condition F0 F : M × [0, T) → Rm, Ft = F(·, t) ∂ ∂t F(p, t) = H(p, t), p ∈ M, t ≥ 0; F(·, 0) = F0 (MCF) Mt = F(M, t), 0 ≤ t < T < ∞

Mean curvature flow (MCF)

F0 : Mn → Rm smooth immersion, M compact MCF with initial condition F0 F : M × [0, T) → Rm, Ft = F(·, t) ∂ ∂t F(p, t) = H(p, t), p ∈ M, t ≥ 0; F(·, 0) = F0 (MCF) Mt = F(M, t), 0 ≤ t < T < ∞

Classical results

[Gage-Hamilton, J. Diff. Geom. 1986], [Grayson, J. Diff. Geom. 1987] C0 closed simple planar curve Then, under (MCF):curve shortening flow, C0 becomes convex and shrinks to a point with round limiting shape

Classical results

[Gage-Hamilton, J. Diff. Geom. 1986], [Grayson, J. Diff. Geom. 1987] C0 closed simple planar curve Then, under (MCF):curve shortening flow, C0 becomes convex and shrinks to a point with round limiting shape [Huisken, J. Diff. Geom. 1984] F0 : Mn → Rn+1, M compact uniformly convex Then, under (MCF), F0 shrinks down to a round point after a finite time

Classical results

[Gage-Hamilton, J. Diff. Geom. 1986], [Grayson, J. Diff. Geom. 1987] C0 closed simple planar curve Then, under (MCF):curve shortening flow, C0 becomes convex and shrinks to a point with round limiting shape [Huisken, J. Diff. Geom. 1984] F0 : Mn → Rn+1, M compact uniformly convex Then, under (MCF), F0 shrinks down to a round point after a finite time [Huisken, J. Diff. Geom. 1990] F : M × [0, T) → Rm solution of (MCF), Mt = F(M, t) lim supt→T maxMt |σ|2 = ∞

Classical results

[Gage-Hamilton, J. Diff. Geom. 1986], [Grayson, J. Diff. Geom. 1987] C0 closed simple planar curve Then, under (MCF):curve shortening flow, C0 becomes convex and shrinks to a point with round limiting shape [Huisken, J. Diff. Geom. 1984] F0 : Mn → Rn+1, M compact uniformly convex Then, under (MCF), F0 shrinks down to a round point after a finite time [Huisken, J. Diff. Geom. 1990] F : M × [0, T) → Rm solution of (MCF), Mt = F(M, t) lim supt→T maxMt |σ|2 = ∞ The so-called Type I singularities (maxMt |σ|2 ≈

c T−t ) look like

self-similar contracting solutions after an appropriate rescaling

Singularities and soliton solutions for MCF

Singularities and soliton solutions for MCF

⊠ Type I singularities model on self-similar solutions for (MCF)

Singularities and soliton solutions for MCF

⊠ Type I singularities model on self-similar solutions for (MCF) φ : Mn → Rm immersion φ self-similar solution for MCF if H = ε φ⊥, ε = ±1

Singularities and soliton solutions for MCF

⊠ Type I singularities model on self-similar solutions for (MCF) φ : Mn → Rm immersion φ self-similar solution for MCF if H = ε φ⊥, ε = ±1 φ self-shrinker if H = −φ⊥ Ft(p) = √1 − 2t φ(p), 0 ≤ t < 1/2, solution of (MCF)

Singularities and soliton solutions for MCF

⊠ Type I singularities model on self-similar solutions for (MCF) φ : Mn → Rm immersion φ self-similar solution for MCF if H = ε φ⊥, ε = ±1 φ self-shrinker if H = −φ⊥ Ft(p) = √1 − 2t φ(p), 0 ≤ t < 1/2, solution of (MCF) φ self-expander if H = φ⊥ Ft(p) = √1 + 2t φ(p), t ≥ 0, solution of (MCF)

Singularities and soliton solutions for MCF

⊠ Type I singularities model on self-similar solutions for (MCF) φ : Mn → Rm immersion φ self-similar solution for MCF if H = ε φ⊥, ε = ±1 φ self-shrinker if H = −φ⊥ Ft(p) = √1 − 2t φ(p), 0 ≤ t < 1/2, solution of (MCF) φ self-expander if H = φ⊥ Ft(p) = √1 + 2t φ(p), t ≥ 0, solution of (MCF) ⊠ Examples of Type II singularities: translating solitons for (MCF)

Singularities and soliton solutions for MCF

⊠ Type I singularities model on self-similar solutions for (MCF) φ : Mn → Rm immersion φ self-similar solution for MCF if H = ε φ⊥, ε = ±1 φ self-shrinker if H = −φ⊥ Ft(p) = √1 − 2t φ(p), 0 ≤ t < 1/2, solution of (MCF) φ self-expander if H = φ⊥ Ft(p) = √1 + 2t φ(p), t ≥ 0, solution of (MCF) ⊠ Examples of Type II singularities: translating solitons for (MCF) φ : Mn → Rm immersion φ translating soliton for MCF if H = e⊥, e ∈ Rm Ft(p) = φ(p) + t e, t ∈ R, solution of (MCF)

Lagrangian submanifolds in Cn

Lagrangian submanifolds in Cn

(R2n ≡ Cn, , , J ) complex Euclidean space ω(., .) = J., . = 1

2dλ

(, ) = , − i ω

Lagrangian submanifolds in Cn

(R2n ≡ Cn, , , J ) complex Euclidean space ω(., .) = J., . = 1

2dλ

(, ) = , − i ω φ : Mn → Cn Lagrangian if φ∗ω = 0 ⇔ J : TM ∼ = T ⊥M σ(v, w) = JAJvw, C(·, ·, ·) = σ(·, ·), J· symmetric Mean curvature form (Maslov form): αH = H ω = JH, . dαH = 0

Lagrangian submanifolds in Cn

(R2n ≡ Cn, , , J ) complex Euclidean space ω(., .) = J., . = 1

2dλ

(, ) = , − i ω φ : Mn → Cn Lagrangian if φ∗ω = 0 ⇔ J : TM ∼ = T ⊥M σ(v, w) = JAJvw, C(·, ·, ·) = σ(·, ·), J· symmetric Mean curvature form (Maslov form): αH = H ω = JH, . dαH = 0 φ∗(dz1 ∧ · · · ∧ dzn) = eiβvolM, β : M → R/2πZ Lagrangian angle map H = J∇β, αH = −dβ

Lagrangian submanifolds in Cn

(R2n ≡ Cn, , , J ) complex Euclidean space ω(., .) = J., . = 1

2dλ

(, ) = , − i ω φ : Mn → Cn Lagrangian if φ∗ω = 0 ⇔ J : TM ∼ = T ⊥M σ(v, w) = JAJvw, C(·, ·, ·) = σ(·, ·), J· symmetric Mean curvature form (Maslov form): αH = H ω = JH, . dαH = 0 φ∗(dz1 ∧ · · · ∧ dzn) = eiβvolM, β : M → R/2πZ Lagrangian angle map H = J∇β, αH = −dβ φ zero Maslov class if β : M → R globally defined φ almost calibrated if cos β ≥ ǫ > 0 φ monotone if [φ∗λ] = c[dβ], c > 0

Hamiltonian stationary Lagrangians (HSL)

Hamiltonian stationary Lagrangians (HSL)

φ : Mn → Cn LAGRANGIAN [Oh, Math. Z. 1993] F : M × (−ǫ, ǫ) → Cn Hamiltonian variation of φ if V = F∗( ∂

∂t ) |t=0 Hamiltonian : αV = JV , . exact

Critical points of volume for Hamiltonian variations: HAMILTONIAN STATIONARY

Hamiltonian stationary Lagrangians (HSL)

φ : Mn → Cn LAGRANGIAN [Oh, Math. Z. 1993] F : M × (−ǫ, ǫ) → Cn Hamiltonian variation of φ if V = F∗( ∂

∂t ) |t=0 Hamiltonian : αV = JV , . exact

Critical points of volume for Hamiltonian variations: HAMILTONIAN STATIONARY φ HSL ⇔ δαH = 0 ⇔ div JH = 0 ⇔ ∆β = 0

Lagrangian MCF

[Smoczyk, 1996] φ = F0 : Mn → R2n ≡ Cn Lagrangian ⇒ Ft, solution to (MCF), Lagrangian, ∀t ∈ [0, T)

Lagrangian MCF

[Smoczyk, 1996] φ = F0 : Mn → R2n ≡ Cn Lagrangian ⇒ Ft, solution to (MCF), Lagrangian, ∀t ∈ [0, T) Study of singularities: [Smoczyk; Calc. Var. PDE 1999, Math. Z. 2002, Calc. Var. PDE 2004] [Wang, J. Diff. Geom. 2001] [Smoczyk & Wang, J. Diff. Geom. 2002] [Thomas & Yau, Comm. Anal. Geom. 2002] [Schoen & Wolfson, 2003] [Chen & Li, Invent. Math. 2004] [Neves; Invent. Math. 2007, Math. Res. Lett. 2010, Ann.of Math. 2013] [Groh, Schwarz, Smoczyk & Zehmisch, Math. Z. 2007]

Lagrangian MCF

[Smoczyk, 1996] φ = F0 : Mn → R2n ≡ Cn Lagrangian ⇒ Ft, solution to (MCF), Lagrangian, ∀t ∈ [0, T) Study of singularities: [Smoczyk; Calc. Var. PDE 1999, Math. Z. 2002, Calc. Var. PDE 2004] [Wang, J. Diff. Geom. 2001] [Smoczyk & Wang, J. Diff. Geom. 2002] [Thomas & Yau, Comm. Anal. Geom. 2002] [Schoen & Wolfson, 2003] [Chen & Li, Invent. Math. 2004] [Neves; Invent. Math. 2007, Math. Res. Lett. 2010, Ann.of Math. 2013] [Groh, Schwarz, Smoczyk & Zehmisch, Math. Z. 2007] Preserved classes of Lagrangians: zero Maslov class, almost calibrated, monotone

Soliton solutions for Lagrangian MCF

⊠ Self-similar solutions for Lagrangian MCF: H = ±φ⊥

Soliton solutions for Lagrangian MCF

⊠ Self-similar solutions for Lagrangian MCF: H = ±φ⊥ Examples: self-shrinkers product of circles and lines S1 × R ⊂ C2, S1 × S1 ⊂ C2

Soliton solutions for Lagrangian MCF

⊠ Self-similar solutions for Lagrangian MCF: H = ±φ⊥ Examples: self-shrinkers product of circles and lines S1 × R ⊂ C2, S1 × S1 ⊂ C2 [Anciaux, 2006] Construction of Lagrangian self-similar solutions to the mean curvature flow in Cn. Geom. Dedicata 120 (2006), 37–48.

Soliton solutions for Lagrangian MCF

⊠ Self-similar solutions for Lagrangian MCF: H = ±φ⊥ Examples: self-shrinkers product of circles and lines S1 × R ⊂ C2, S1 × S1 ⊂ C2 [Anciaux, 2006] Construction of Lagrangian self-similar solutions to the mean curvature flow in Cn. Geom. Dedicata 120 (2006), 37–48. [Lee & Wang, 2009] Hamiltonian stationary shrinkers and expanders for Lagrangian mean curvature flows. J. Differential Geom. 83 (2009), 27–42. [Lee & Wang, 2010] Hamiltonian stationary cones and self-similar solutions in higher dimension. Trans. Amer. Math. Soc. 362 (2010), 1491–1503.

Soliton solutions for Lagrangian MCF

⊠ Self-similar solutions for Lagrangian MCF: H = ±φ⊥ Examples: self-shrinkers product of circles and lines S1 × R ⊂ C2, S1 × S1 ⊂ C2 [Anciaux, 2006] Construction of Lagrangian self-similar solutions to the mean curvature flow in Cn. Geom. Dedicata 120 (2006), 37–48. [Lee & Wang, 2009] Hamiltonian stationary shrinkers and expanders for Lagrangian mean curvature flows. J. Differential Geom. 83 (2009), 27–42. [Lee & Wang, 2010] Hamiltonian stationary cones and self-similar solutions in higher dimension. Trans. Amer. Math. Soc. 362 (2010), 1491–1503. [Joyce, Lee & Tsui, 2010] Self-similar solutions and translating solitons for Lagrangian mean curvature flow. J. Differential Geom. 84 (2010), 127-161.

Soliton solutions for Lagrangian MCF

⊠ Self-similar solutions for Lagrangian MCF: H = ±φ⊥ Examples: self-shrinkers product of circles and lines S1 × R ⊂ C2, S1 × S1 ⊂ C2 [Anciaux, 2006] Construction of Lagrangian self-similar solutions to the mean curvature flow in Cn. Geom. Dedicata 120 (2006), 37–48. [Lee & Wang, 2009] Hamiltonian stationary shrinkers and expanders for Lagrangian mean curvature flows. J. Differential Geom. 83 (2009), 27–42. [Lee & Wang, 2010] Hamiltonian stationary cones and self-similar solutions in higher dimension. Trans. Amer. Math. Soc. 362 (2010), 1491–1503. [Joyce, Lee & Tsui, 2010] Self-similar solutions and translating solitons for Lagrangian mean curvature flow. J. Differential Geom. 84 (2010), 127-161. [Lotay & Neves, 2012] Uniqueness of Lagrangian self-expanders. Preprint 2012

Soliton solutions for Lagrangian MCF

⊠ Translating solitons for Lagrangian MCF: H = e⊥

Soliton solutions for Lagrangian MCF

⊠ Translating solitons for Lagrangian MCF: H = e⊥ β = −φ, Je+ constant

Soliton solutions for Lagrangian MCF

⊠ Translating solitons for Lagrangian MCF: H = e⊥ β = −φ, Je+ constant Simple examples: products with the grim-reaper curve and lines

Soliton solutions for Lagrangian MCF

⊠ Translating solitons for Lagrangian MCF: H = e⊥ β = −φ, Je+ constant Simple examples: products with the grim-reaper curve and lines

Soliton solutions for Lagrangian MCF

⊠ Translating solitons for Lagrangian MCF: H = e⊥ β = −φ, Je+ constant Simple examples: products with the grim-reaper curve and lines [Neves & Tian, 2007] Translating solutions to Lagrangian mean curvature flow. To appear in Trans. Amer. Math. Soc.

Soliton solutions for Lagrangian MCF

⊠ Translating solitons for Lagrangian MCF: H = e⊥ β = −φ, Je+ constant Simple examples: products with the grim-reaper curve and lines [Neves & Tian, 2007] Translating solutions to Lagrangian mean curvature flow. To appear in Trans. Amer. Math. Soc. [Joyce, Lee & Tsui, 2010] Self-similar solutions and translating solitons for Lagrangian mean curvature flow. J. Differential Geom. 84 (2010), 127-161.

SELF-SIMILAR SOLUTIONS

Self-expanders Φδ : R2 → C2, δ > 0

Self-expanders Φδ : R2 → C2, δ > 0

Φδ(s, t) =

• i sδ cosh t e− i s

cδ , tδ sinh t ei cδ s

sδ = sinh δ, cδ = cosh δ, tδ = tanh δ HSL Hδ = Φ⊥

δ

Self-expanders Φδ : R2 → C2, δ > 0

Φδ(s, t) =

• i sδ cosh t e− i s

cδ , tδ sinh t ei cδ s

sδ = sinh δ, cδ = cosh δ, tδ = tanh δ HSL Hδ = Φ⊥

δ

cosh2 δ / ∈ Q, Φδ embedded plane;

Self-expanders Φδ : R2 → C2, δ > 0

Φδ(s, t) =

• i sδ cosh t e− i s

cδ , tδ sinh t ei cδ s

sδ = sinh δ, cδ = cosh δ, tδ = tanh δ HSL Hδ = Φ⊥

δ

cosh2 δ / ∈ Q, Φδ embedded plane; cosh2 δ = p/q ∈ Q, (p, q) = 1 Φp,q : R2 → C2, p > q [Lee & Wang, 2009] Φp,q(s, t) = √p − q i √q cosh t e−i√ q

p s, 1

√p sinh t ei√ p

q s

• ◮ Φp,q(s + 2π√pq, t) = Φp,q(s, t), ∀(s, t) ∈ R2 cylinder

Self-expanders Φδ : R2 → C2, δ > 0

Φδ(s, t) =

• i sδ cosh t e− i s

cδ , tδ sinh t ei cδ s

sδ = sinh δ, cδ = cosh δ, tδ = tanh δ HSL Hδ = Φ⊥

δ

cosh2 δ / ∈ Q, Φδ embedded plane; cosh2 δ = p/q ∈ Q, (p, q) = 1 Φp,q : R2 → C2, p > q [Lee & Wang, 2009] Φp,q(s, t) = √p − q i √q cosh t e−i√ q

p s, 1

√p sinh t ei√ p

q s

• ◮ Φp,q(s + 2π√pq, t) = Φp,q(s, t), ∀(s, t) ∈ R2 cylinder

◮ p odd, q even: Φp,q(s + π√pq, −t) = Φp,q(s, t), ∀(s, t) ∈ R2 Moebius strip

Plane Φδ, cosh2 δ / ∈ Q

Cylinder Φ3,1

Moebius strip Φ3,2

Self-shrinkers Υγ : R2 → C2, 0 < γ < π/2

Self-shrinkers Υγ : R2 → C2, 0 < γ < π/2

Υγ(s, t) =

• −i sγ cosh t e

i s cγ , tγ sinh t e−i cγ s

sγ = sin γ, cγ = cos γ, tγ = tan γ HSL Hγ = − Υ⊥

γ

Self-shrinkers Υγ : R2 → C2, 0 < γ < π/2

Υγ(s, t) =

• −i sγ cosh t e

i s cγ , tγ sinh t e−i cγ s

sγ = sin γ, cγ = cos γ, tγ = tan γ HSL Hγ = − Υ⊥

γ

cos2 γ / ∈ Q, Υγ embedded plane;

Self-shrinkers Υγ : R2 → C2, 0 < γ < π/2

Υγ(s, t) =

• −i sγ cosh t e

i s cγ , tγ sinh t e−i cγ s

sγ = sin γ, cγ = cos γ, tγ = tan γ HSL Hγ = − Υ⊥

γ

cos2 γ / ∈ Q, Υγ embedded plane; cos2 γ = p/q ∈ Q, (p, q) = 1 Υp,q : R2 → C2, p < q [Lee & Wang, 2009] Υp,q(s, t) = √q − p −i √q cosh t ei√ q

p s, 1

√p sinh t e−i√ p

q s

• ◮ Υp,q(s + 2π√pq, t) = Υp,q(s, t), ∀(s, t) ∈ R2 cylinder

Self-shrinkers Υγ : R2 → C2, 0 < γ < π/2

Υγ(s, t) =

• −i sγ cosh t e

i s cγ , tγ sinh t e−i cγ s

sγ = sin γ, cγ = cos γ, tγ = tan γ HSL Hγ = − Υ⊥

γ

cos2 γ / ∈ Q, Υγ embedded plane; cos2 γ = p/q ∈ Q, (p, q) = 1 Υp,q : R2 → C2, p < q [Lee & Wang, 2009] Υp,q(s, t) = √q − p −i √q cosh t ei√ q

p s, 1

√p sinh t e−i√ p

q s

• ◮ Υp,q(s + 2π√pq, t) = Υp,q(s, t), ∀(s, t) ∈ R2 cylinder

◮ q even, p odd: Υp,q(s + π√pq, −t) = Υp,q(s, t), ∀(s, t) ∈ R2 Moebius strip

Plane Υγ, cos2 γ / ∈ Q

Cylinder Υ1,3

Moebius strip Υ1,2

Self-shrinkers Ψν : S1 × R → C2, ν > 0

Self-shrinkers Ψν : S1 × R → C2, ν > 0

Ψν(ei s, t) =

• cν cos s e

i t sν , tν sin s ei sν t

sν = sinh ν, cν = cosh ν, tν = coth ν HSL Hν = −Ψ⊥

ν

Self-shrinkers Ψν : S1 × R → C2, ν > 0

Ψν(ei s, t) =

• cν cos s e

i t sν , tν sin s ei sν t

sν = sinh ν, cν = cosh ν, tν = coth ν HSL Hν = −Ψ⊥

ν

sinh2 ν / ∈ Q, Ψν embedded cylinder;

Self-shrinkers Ψν : S1 × R → C2, ν > 0

Ψν(ei s, t) =

• cν cos s e

i t sν , tν sin s ei sν t

sν = sinh ν, cν = cosh ν, tν = coth ν HSL Hν = −Ψ⊥

ν

sinh2 ν / ∈ Q, Ψν embedded cylinder; sinh2 ν =m/n ∈ Q, (m, n)=1 Ψm,n : S1 × R → C2, (m, n) = 1 [Lee & Wang, 2010] Ψm,n(s, t) = √ m + n 1 √n cos s ei√ n

m t,

1 √m sin s ei√ m

n t

• ◮ Ψm,n(s+2π, t)=Ψm,n(s, t)=Ψm,n(s, t+2π√mn), ∀(s, t)∈R2

Family Ψm,n

Family Ψm,n

◮ m, n odd: Ψm,n(s + π, t + π√mn) = Ψm,n(s, t), ∀(s, t) ∈ R2 Tm,n = Ψm,n(R2/Λm,n) torus ◮ m odd, n even: Ψm,n(2π−s, t+π√mn)=Ψm,n(s, t), ∀(s, t)∈R2 Klein bottle ◮ m even, n odd: Ψm,n(π−s, t+π√mn)=Ψm,n(s, t), ∀(s, t)∈R2 Klein bottle

Family Ψm,n

◮ m, n odd: Ψm,n(s + π, t + π√mn) = Ψm,n(s, t), ∀(s, t) ∈ R2 Tm,n = Ψm,n(R2/Λm,n) torus ◮ m odd, n even: Ψm,n(2π−s, t+π√mn)=Ψm,n(s, t), ∀(s, t)∈R2 Klein bottle ◮ m even, n odd: Ψm,n(π−s, t+π√mn)=Ψm,n(s, t), ∀(s, t)∈R2 Klein bottle Clifford torus T1,1 only one embedded

Family Ψm,n

◮ m, n odd: Ψm,n(s + π, t + π√mn) = Ψm,n(s, t), ∀(s, t) ∈ R2 Tm,n = Ψm,n(R2/Λm,n) torus ◮ m odd, n even: Ψm,n(2π−s, t+π√mn)=Ψm,n(s, t), ∀(s, t)∈R2 Klein bottle ◮ m even, n odd: Ψm,n(π−s, t+π√mn)=Ψm,n(s, t), ∀(s, t)∈R2 Klein bottle Clifford torus T1,1 only one embedded Willmore(Tm,n) = 2Area(Tm,n) =     

4(m+n)2π2 √mn

, m or n even

2(m+n)2π2 √mn

, m and n odd

Clifford torus

Torus Ψ1,3

Klein bottle Ψ1,2

Classification results

Classification results

Theorem φ : M2 → C2 HSL self-similar solution for MCF (a) φ self-expander (H = φ⊥) ⇒ φ

loc

∼ Φδ : R2 → C2, δ > 0 (b) φ self-shrinker (H = −φ⊥) ⇒ φ

loc

∼

(i) S1 × R (ii) S1 × S1 (iii) Υγ : R2 → C2, 0 < γ < π/2 (iv) Ψν : S1 × R → C2, ν > 0

Classification results

Theorem φ : M2 → C2 HSL self-similar solution for MCF (a) φ self-expander (H = φ⊥) ⇒ φ

loc

∼ Φδ : R2 → C2, δ > 0 (b) φ self-shrinker (H = −φ⊥) ⇒ φ

loc

∼

(i) S1 × R (ii) S1 × S1 (iii) Υγ : R2 → C2, 0 < γ < π/2 (iv) Ψν : S1 × R → C2, ν > 0

Corollary φ : M → C2 HSL self-similar solution for MCF M compact orientable ⇒ φ(M) ∼ Tm,n

THE CLIFFORD TORUS AS A SELF-SHRINKER

Self-shrinkers: notion and examples

Self-shrinkers: notion and examples

φ : Mn → Rm self-shrinker if H = −φ⊥ (H = trace σ)

Self-shrinkers: notion and examples

φ : Mn → Rm self-shrinker if H = −φ⊥ (H = trace σ) ◮ F(p, t) = √1 − 2t Φ(p), 0 ≤ t < 1/2, solution to (MCF)

Self-shrinkers: notion and examples

φ : Mn → Rm self-shrinker if H = −φ⊥ (H = trace σ) ◮ F(p, t) = √1 − 2t Φ(p), 0 ≤ t < 1/2, solution to (MCF) Example (Sphere) Sn(√n) ֒ → Rn+1, |σ|2 ≡ 1

Self-shrinkers: notion and examples

φ : Mn → Rm self-shrinker if H = −φ⊥ (H = trace σ) ◮ F(p, t) = √1 − 2t Φ(p), 0 ≤ t < 1/2, solution to (MCF) Example (Sphere) Sn(√n) ֒ → Rn+1, |σ|2 ≡ 1 Example (Clifford) Sn1(√n1) × Sn2(√n2) ֒ → Rn+2, n1, n2 ∈ N, n1 + n2 = n, |σ|2 ≡ 2

Self-shrinkers: notion and examples

φ : Mn → Rm self-shrinker if H = −φ⊥ (H = trace σ) ◮ F(p, t) = √1 − 2t Φ(p), 0 ≤ t < 1/2, solution to (MCF) Example (Sphere) Sn(√n) ֒ → Rn+1, |σ|2 ≡ 1 Example (Clifford) Sn1(√n1) × Sn2(√n2) ֒ → Rn+2, n1, n2 ∈ N, n1 + n2 = n, |σ|2 ≡ 2 Example (Product of n-circles) S1×

n)

· · · ×S1 ֒ → R2n, |σ|2 ≡ n, Lagrangian in R2n ≡ Cn

Self-shrinkers: notion and examples

φ : Mn → Rm self-shrinker if H = −φ⊥ (H = trace σ) ◮ F(p, t) = √1 − 2t Φ(p), 0 ≤ t < 1/2, solution to (MCF) Example (Sphere) Sn(√n) ֒ → Rn+1, |σ|2 ≡ 1 Example (Clifford) Sn1(√n1) × Sn2(√n2) ֒ → Rn+2, n1, n2 ∈ N, n1 + n2 = n, |σ|2 ≡ 2 Example (Product of n-circles) S1×

n)

· · · ×S1 ֒ → R2n, |σ|2 ≡ n, Lagrangian in R2n ≡ Cn Example (Product of a circle and (n−1)-sphere) S1 × Sn−1 → Cn ≡ R2n, (eit, (x1, . . . , xn)) → √n eit (x1, . . . , xn) |σ|2 ≡ (3n − 2)/n, Lagrangian in Cn ≡ R2n

Case n = 1: Self-shrinking curves

Case n = 1: Self-shrinking curves

− → κα = −α⊥

Case n = 1: Self-shrinking curves

− → κα = −α⊥ [Abresh & Langer, J. Diff. Geom. 1986] curves

Classification and rigidity results, n ≥ 2

Classification and rigidity results, n ≥ 2

[Huisken, J. Diff. Geom. 1990] φ : Mn → Rn+1, M compact self-shrinker H ≥ 0 ⇒ Mn ≡ Sn(√n)

Classification and rigidity results, n ≥ 2

[Huisken, J. Diff. Geom. 1990] φ : Mn → Rn+1, M compact self-shrinker H ≥ 0 ⇒ Mn ≡ Sn(√n) [Smoczyk, Int. Math. Res. Not. 2005] φ : Mn → Rm, M compact self-shrinker |H| > 0; ∇⊥ν = 0, ν = H/|H|

• Mn spherical: Mn → Sm−1(√n), ˆ

H = 0

Classification and rigidity results, n ≥ 2

[Huisken, J. Diff. Geom. 1990] φ : Mn → Rn+1, M compact self-shrinker H ≥ 0 ⇒ Mn ≡ Sn(√n) [Smoczyk, Int. Math. Res. Not. 2005] φ : Mn → Rm, M compact self-shrinker |H| > 0; ∇⊥ν = 0, ν = H/|H|

• Mn spherical: Mn → Sm−1(√n), ˆ

H = 0 [Le & Sesum, Comm. Anal. Geom. 2011] [Cao & Li, Calc. Var. PDE 2012] φ : Mn → Rm, M compact self-shrinker |σ|2 ≤ 1 ⇒ |σ|2 ≡ 1, Mn ≡ Sn(√n) ⊂ Rn+1

Classification and rigidity results, n ≥ 2

[Li & Wei, preprint 2012] φ : Mn → Rn+2, M compact self-shrinker

Classification and rigidity results, n ≥ 2

[Li & Wei, preprint 2012] φ : Mn → Rn+2, M compact self-shrinker |H| > 0; ∇⊥ν = 0, ν = H/|H| 1 ≤ |σ|2 ≤ 2

Classification and rigidity results, n ≥ 2

[Li & Wei, preprint 2012] φ : Mn → Rn+2, M compact self-shrinker |H| > 0; ∇⊥ν = 0, ν = H/|H| 1 ≤ |σ|2 ≤ 2

1 either |σ|2 ≡ 1, M ≡ Sn(√n) ⊂ Rn+1 2 or |σ|2 ≡ 2, M ≡ Sn1(√n1) × Sn2(√n2) ⊂ Rn+2, n1 + n2 = n

Classification and rigidity results, n ≥ 2

[Li & Wei, preprint 2012] φ : Mn → Rn+2, M compact self-shrinker |H| > 0; ∇⊥ν = 0, ν = H/|H| 1 ≤ |σ|2 ≤ 2

1 either |σ|2 ≡ 1, M ≡ Sn(√n) ⊂ Rn+1 2 or |σ|2 ≡ 2, M ≡ Sn1(√n1) × Sn2(√n2) ⊂ Rn+2, n1 + n2 = n

φ : M2 → R2+p, M compact self-shrinker |H| > 0; ∇⊥ν = 0, ν = H/|H| 1 ≤ |σ|2 ≤ 5

3

Classification and rigidity results, n ≥ 2

[Li & Wei, preprint 2012] φ : Mn → Rn+2, M compact self-shrinker |H| > 0; ∇⊥ν = 0, ν = H/|H| 1 ≤ |σ|2 ≤ 2

1 either |σ|2 ≡ 1, M ≡ Sn(√n) ⊂ Rn+1 2 or |σ|2 ≡ 2, M ≡ Sn1(√n1) × Sn2(√n2) ⊂ Rn+2, n1 + n2 = n

φ : M2 → R2+p, M compact self-shrinker |H| > 0; ∇⊥ν = 0, ν = H/|H| 1 ≤ |σ|2 ≤ 5

3 1 either |σ|2 ≡ 1, M ≡ S2(

√ 2) ⊂ R3

2 or |σ|2 ≡ 5/3, M ≡ S2(

√ 6) → S4( √ 2) ֒ → R5 Veronese

Classification and rigidity results, n ≥ 2

[Li & Wei, preprint 2012] φ : Mn → Rn+2, M compact self-shrinker |H| > 0; ∇⊥ν = 0, ν = H/|H| 1 ≤ |σ|2 ≤ 2

1 either |σ|2 ≡ 1, M ≡ Sn(√n) ⊂ Rn+1 2 or |σ|2 ≡ 2, M ≡ Sn1(√n1) × Sn2(√n2) ⊂ Rn+2, n1 + n2 = n

φ : M2 → R2+p, M compact self-shrinker |H| > 0; ∇⊥ν = 0, ν = H/|H| 1 ≤ |σ|2 ≤ 5

3 1 either |σ|2 ≡ 1, M ≡ S2(

√ 2) ⊂ R3

2 or |σ|2 ≡ 5/3, M ≡ S2(

√ 6) → S4( √ 2) ֒ → R5 Veronese

|H| > 0; ∇⊥ν = 0, ν = H/|H|

5 3 ≤ |σ|2 ≤ 11 6

Classification and rigidity results, n ≥ 2

[Li & Wei, preprint 2012] φ : Mn → Rn+2, M compact self-shrinker |H| > 0; ∇⊥ν = 0, ν = H/|H| 1 ≤ |σ|2 ≤ 2

1 either |σ|2 ≡ 1, M ≡ Sn(√n) ⊂ Rn+1 2 or |σ|2 ≡ 2, M ≡ Sn1(√n1) × Sn2(√n2) ⊂ Rn+2, n1 + n2 = n

φ : M2 → R2+p, M compact self-shrinker |H| > 0; ∇⊥ν = 0, ν = H/|H| 1 ≤ |σ|2 ≤ 5

3 1 either |σ|2 ≡ 1, M ≡ S2(

√ 2) ⊂ R3

2 or |σ|2 ≡ 5/3, M ≡ S2(

√ 6) → S4( √ 2) ֒ → R5 Veronese

|H| > 0; ∇⊥ν = 0, ν = H/|H|

5 3 ≤ |σ|2 ≤ 11 6 1 either |σ|2 ≡ 5/3, M ≡ S2(

√ 6) → S4( √ 2) ֒ → R5 Veronese

2 or |σ|2 ≡ 11/6, M ≡ S2(

√ 12) → S6( √ 2) ֒ → R7 Boruvka

Our contribution (arbitrary dimension and codimension)

Theorem A φ : Mn → Rn+p, M compact self-shrinker |H|2 constant or |H|2 ≤ n or |H|2 ≥ n |σ|2 ≤ 3p−4

2p−3

Our contribution (arbitrary dimension and codimension)

Theorem A φ : Mn → Rn+p, M compact self-shrinker |H|2 constant or |H|2 ≤ n or |H|2 ≥ n |σ|2 ≤ 3p−4

2p−3

Then:

1 either |σ|2 ≡ 1, M ≡ Sn(√n) ⊂ Rn+1 [p = 1]

Our contribution (arbitrary dimension and codimension)

Theorem A φ : Mn → Rn+p, M compact self-shrinker |H|2 constant or |H|2 ≤ n or |H|2 ≥ n |σ|2 ≤ 3p−4

2p−3

Then:

1 either |σ|2 ≡ 1, M ≡ Sn(√n) ⊂ Rn+1 [p = 1] 2 or |σ|2 ≡ 3p−4 2p−3,

(i) either M ≡ Sn1(√n1) × Sn2(√n2) ⊂ Rn+2, n1 + n2 = n, (with |σ|2 ≡ 2) [p = 2] (ii) or Veronese M ≡ S2( √ 6) ⊂ R5 (with |σ|2 ≡ 5/3) [n = 2, p = 3]

Proof of Theorem A

Proof of Theorem A

H = −φ⊥ ⇒ △|φ|2 = 2(n − |H|2)

Proof of Theorem A

H = −φ⊥ ⇒ △|φ|2 = 2(n − |H|2) ⇒ 0 =

• M(n − |H|2)dµ

Proof of Theorem A

H = −φ⊥ ⇒ △|φ|2 = 2(n − |H|2) ⇒ 0 =

• M(n − |H|2)dµ

[|H|2 constant or |H|2 ≤ n or |H|2 ≥ n] ⇒ |H|2 ≡ n

Proof of Theorem A

H = −φ⊥ ⇒ △|φ|2 = 2(n − |H|2) ⇒ 0 =

• M(n − |H|2)dµ

[|H|2 constant or |H|2 ≤ n or |H|2 ≥ n] ⇒ |H|2 ≡ n ⇒ |φ|2 ≡ n φ : Mn → Sn+p−1(√n), ˆ H ≡ 0,

Proof of Theorem A

H = −φ⊥ ⇒ △|φ|2 = 2(n − |H|2) ⇒ 0 =

• M(n − |H|2)dµ

[|H|2 constant or |H|2 ≤ n or |H|2 ≥ n] ⇒ |H|2 ≡ n ⇒ |φ|2 ≡ n φ : Mn → Sn+p−1(√n), ˆ H ≡ 0, |σ|2 = 1 + |ˆ σ|2,

Proof of Theorem A

H = −φ⊥ ⇒ △|φ|2 = 2(n − |H|2) ⇒ 0 =

• M(n − |H|2)dµ

[|H|2 constant or |H|2 ≤ n or |H|2 ≥ n] ⇒ |H|2 ≡ n ⇒ |φ|2 ≡ n φ : Mn → Sn+p−1(√n), ˆ H ≡ 0, |σ|2 = 1 + |ˆ σ|2, |ˆ σ|2 ≤

p−1 2p−3

Proof of Theorem A

H = −φ⊥ ⇒ △|φ|2 = 2(n − |H|2) ⇒ 0 =

• M(n − |H|2)dµ

[|H|2 constant or |H|2 ≤ n or |H|2 ≥ n] ⇒ |H|2 ≡ n ⇒ |φ|2 ≡ n φ : Mn → Sn+p−1(√n), ˆ H ≡ 0, |σ|2 = 1 + |ˆ σ|2, |ˆ σ|2 ≤

p−1 2p−3

[Simons, 1968] [Lawson, 1969] [Chern, DoCarmo & Kobayashi, 1978] Mn compact minimal submanifold in Sn+q, |ˆ σ|2 ≤

n 2−1/q

Then either |ˆ σ|2 ≡ 0, Mn ≡ Sn, or |ˆ σ|2 ≡

n 2−1/q,

Mn ≡ Sk(

• k

n) × Sn−k(

• n−k

n ) ⊂ Sn+1, 1≤k ≤n−1,

• r Veronese S2(

√ 3) ֒ → S4

Proof of Theorem A

H = −φ⊥ ⇒ △|φ|2 = 2(n − |H|2) ⇒ 0 =

• M(n − |H|2)dµ

[|H|2 constant or |H|2 ≤ n or |H|2 ≥ n] ⇒ |H|2 ≡ n ⇒ |φ|2 ≡ n φ : Mn → Sn+p−1(√n), ˆ H ≡ 0, |σ|2 = 1 + |ˆ σ|2, |ˆ σ|2 ≤

p−1 2p−3

[Simons, 1968] [Lawson, 1969] [Chern, DoCarmo & Kobayashi, 1978] Mn compact minimal submanifold in Sn+q, |ˆ σ|2 ≤

n 2−1/q

Then either |ˆ σ|2 ≡ 0, Mn ≡ Sn, or |ˆ σ|2 ≡

n 2−1/q,

Mn ≡ Sk(

• k

n) × Sn−k(

• n−k

n ) ⊂ Sn+1, 1≤k ≤n−1,

• r Veronese S2(

√ 3) ֒ → S4 Take a dilation in Rn+q+1 ⊃ Sn+q of ratio √n and put q = p − 1

Our contribution (codimension n ≥ 2)

Corollary A φ : Mn → R2n, M compact self-shrinker (codimension n ≥ 2) |H|2 constant or |H|2 ≤ n or |H|2 ≥ n |σ|2 ≤ 3n−4

2n−3

Our contribution (codimension n ≥ 2)

Corollary A φ : Mn → R2n, M compact self-shrinker (codimension n ≥ 2) |H|2 constant or |H|2 ≤ n or |H|2 ≥ n |σ|2 ≤ 3n−4

2n−3

Then n = 2, |σ|2 ≡ 2, M ≡ S1 × S1 ⊂ R4 Clifford torus

Our contribution (codimension n ≥ 2)

Corollary A φ : Mn → R2n, M compact self-shrinker (codimension n ≥ 2) |H|2 constant or |H|2 ≤ n or |H|2 ≥ n |σ|2 ≤ 3n−4

2n−3

Then n = 2, |σ|2 ≡ 2, M ≡ S1 × S1 ⊂ R4 Clifford torus Self-shrinking tori in R4: (i) Abresch-Langer tori, product of two Abresch-Langer curves (ii) Anciaux tori [Anciaux, Geom. Dedicata 2006] (iii) Lee-Wang tori [Lee & Wang, Trans. Amer. Math. Soc. 2010] (iv) Lawson tori [Lawson, Ann. of Math. 1970]

Our contribution in the Lagrangian setting I

Theorem φ : M2 → R4 compact Lagrangian self-shrinker |H|2 constant or |H|2 ≤ 2 or |H|2 ≥ 2 Then M2 ≡ S1 × S1

Our contribution in the Lagrangian setting I

Theorem φ : M2 → R4 compact Lagrangian self-shrinker |H|2 constant or |H|2 ≤ 2 or |H|2 ≥ 2 Then M2 ≡ S1 × S1 Proof:

Our contribution in the Lagrangian setting I

Theorem φ : M2 → R4 compact Lagrangian self-shrinker |H|2 constant or |H|2 ≤ 2 or |H|2 ≥ 2 Then M2 ≡ S1 × S1 Proof: Hypothesis on H ⇒ |φ|2 = |H|2 ≡ 2

Our contribution in the Lagrangian setting I

Theorem φ : M2 → R4 compact Lagrangian self-shrinker |H|2 constant or |H|2 ≤ 2 or |H|2 ≥ 2 Then M2 ≡ S1 × S1 Proof: Hypothesis on H ⇒ |φ|2 = |H|2 ≡ 2

Smoczyk

⇒ ∇⊥H = 0

φ Lagr.

⇒ JH parallel

Our contribution in the Lagrangian setting I

Theorem φ : M2 → R4 compact Lagrangian self-shrinker |H|2 constant or |H|2 ≤ 2 or |H|2 ≥ 2 Then M2 ≡ S1 × S1 Proof: Hypothesis on H ⇒ |φ|2 = |H|2 ≡ 2

Smoczyk

⇒ ∇⊥H = 0

φ Lagr.

⇒ JH parallel

Urbano

⇒ M2 ≡ S1(r1) × S1(r2)

H=−φ

⇒ M2 ≡ S1 × S1

Our contribution in the Lagrangian setting II

Theorem φ : M2 → R4 compact orientable Lagrangian self-shrinker |σ|2 ≤ 2 Then |σ|2 ≡ 2, M torus

Our contribution in the Lagrangian setting II

Theorem φ : M2 → R4 compact orientable Lagrangian self-shrinker |σ|2 ≤ 2 Then |σ|2 ≡ 2, M torus If, in addition, K ≥ 0 or K ≤ 0, then M2 ≡ S1 × S1

Our contribution in the Lagrangian setting II

Theorem φ : M2 → R4 compact orientable Lagrangian self-shrinker |σ|2 ≤ 2 Then |σ|2 ≡ 2, M torus If, in addition, K ≥ 0 or K ≤ 0, then M2 ≡ S1 × S1 Proof: △|φ|2 = 2(2 − |H|2) ⇒

• M |H|2dµ = 2 Area(M)

Our contribution in the Lagrangian setting II

Theorem φ : M2 → R4 compact orientable Lagrangian self-shrinker |σ|2 ≤ 2 Then |σ|2 ≡ 2, M torus If, in addition, K ≥ 0 or K ≤ 0, then M2 ≡ S1 × S1 Proof: △|φ|2 = 2(2 − |H|2) ⇒

• M |H|2dµ = 2 Area(M)

Gauss equation: 2K = |H|2 − |σ|2

Our contribution in the Lagrangian setting II

Theorem φ : M2 → R4 compact orientable Lagrangian self-shrinker |σ|2 ≤ 2 Then |σ|2 ≡ 2, M torus If, in addition, K ≥ 0 or K ≤ 0, then M2 ≡ S1 × S1 Proof: △|φ|2 = 2(2 − |H|2) ⇒

• M |H|2dµ = 2 Area(M)

Gauss equation: 2K = |H|2 − |σ|2 Gauss-Bonnet: 8π(1 − gen(M)) =

• M(|H|2 − |σ|2)dµ =
• M(2 − |σ|2)dµ

Our contribution in the Lagrangian setting II

Theorem φ : M2 → R4 compact orientable Lagrangian self-shrinker |σ|2 ≤ 2 Then |σ|2 ≡ 2, M torus If, in addition, K ≥ 0 or K ≤ 0, then M2 ≡ S1 × S1 Proof: △|φ|2 = 2(2 − |H|2) ⇒

• M |H|2dµ = 2 Area(M)

Gauss equation: 2K = |H|2 − |σ|2 Gauss-Bonnet: 0 ≥ 8π(1 − gen(M)) =

• M(|H|2 − |σ|2)dµ =
• M(2 − |σ|2)dµ

[Smoczyk, 2000] M can not be a sphere

Our contribution in the Lagrangian setting II

Theorem φ : M2 → R4 compact orientable Lagrangian self-shrinker |σ|2 ≤ 2 Then |σ|2 ≡ 2, M torus If, in addition, K ≥ 0 or K ≤ 0, then M2 ≡ S1 × S1 Proof: △|φ|2 = 2(2 − |H|2) ⇒

• M |H|2dµ = 2 Area(M)

Gauss equation: 2K = |H|2 − |σ|2 Gauss-Bonnet: 0 ≥ 8π(1 − gen(M)) =

• M(|H|2 − |σ|2)dµ =
• M(2 − |σ|2)dµ

[Smoczyk, 2000] M can not be a sphere |σ|2 ≤ 2

Our contribution in the Lagrangian setting II

Theorem φ : M2 → R4 compact orientable Lagrangian self-shrinker |σ|2 ≤ 2 Then |σ|2 ≡ 2, M torus If, in addition, K ≥ 0 or K ≤ 0, then M2 ≡ S1 × S1 Proof: △|φ|2 = 2(2 − |H|2) ⇒

• M |H|2dµ = 2 Area(M)

Gauss equation: 2K = |H|2 − |σ|2 Gauss-Bonnet: 0 ≥ 8π(1 − gen(M)) =

• M(|H|2 − |σ|2)dµ =
• M(2 − |σ|2)dµ≥ 0

[Smoczyk, 2000] M can not be a sphere |σ|2 ≤ 2 ⇒ M torus, |σ|2 ≡ 2

Our contribution in the Lagrangian setting II

Theorem φ : M2 → R4 compact orientable Lagrangian self-shrinker |σ|2 ≤ 2 Then |σ|2 ≡ 2, M torus If, in addition, K ≥ 0 or K ≤ 0, then M2 ≡ S1 × S1 Proof: △|φ|2 = 2(2 − |H|2) ⇒

• M |H|2dµ = 2 Area(M)

Gauss equation: 2K = |H|2 − |σ|2 Gauss-Bonnet: 0 ≥ 8π(1 − gen(M)) =

• M(|H|2 − |σ|2)dµ =
• M(2 − |σ|2)dµ≥ 0

[Smoczyk, 2000] M can not be a sphere |σ|2 ≤ 2 ⇒ M torus, |σ|2 ≡ 2 (2K = |H|2 − 2) K ≥ 0 or K ≤ 0 ⇒ |H|2 ≥ 2 or |H|2 ≤ 2 ⇒ M Clifford

Our contribution in the Lagrangian setting III

Theorem φ : M2 → R4 compact self-shrinker φ Hamiltonian stationary Lagrangian embedding Then M2 ≡ S1 × S1

Our contribution in the Lagrangian setting III

Theorem φ : M2 → R4 compact self-shrinker φ Hamiltonian stationary Lagrangian embedding Then M2 ≡ S1 × S1 Proof: [Castro & Lerma, 2010] Lee-Wang tori Tm,n only compact orientable Hamiltonian stationary Lagrangian self-shrinkers

Our contribution in the Lagrangian setting III

Theorem φ : M2 → R4 compact self-shrinker φ Hamiltonian stationary Lagrangian embedding Then M2 ≡ S1 × S1 Proof: [Castro & Lerma, 2010] Lee-Wang tori Tm,n only compact orientable Hamiltonian stationary Lagrangian self-shrinkers Tm,n ≡ Ψm,n : R2 → C2, m, n ∈ N, (m, n) = 1, m ≤ n Ψm,n(s, t) = √m + n

• 1

√n cos s ei√ n

m t,

1 √m sin s ei√ m

n t

Our contribution in the Lagrangian setting III

Theorem φ : M2 → R4 compact self-shrinker φ Hamiltonian stationary Lagrangian embedding Then M2 ≡ S1 × S1 Proof: [Castro & Lerma, 2010] Lee-Wang tori Tm,n only compact orientable Hamiltonian stationary Lagrangian self-shrinkers Tm,n ≡ Ψm,n : R2 → C2, m, n ∈ N, (m, n) = 1, m ≤ n Ψm,n(s, t) = √m + n

• 1

√n cos s ei√ n

m t,

1 √m sin s ei√ m

n t

Clifford torus T1,1 ≡ Ψ1,1(s, t) = √ 2eit(cos s, sin s ) only embedded

Our contribution in the Lagrangian setting III

Theorem φ : M2 → R4 compact self-shrinker φ Hamiltonian stationary Lagrangian embedding Then M2 ≡ S1 × S1 Proof: [Castro & Lerma, 2010] Lee-Wang tori Tm,n only compact orientable Hamiltonian stationary Lagrangian self-shrinkers Tm,n ≡ Ψm,n : R2 → C2, m, n ∈ N, (m, n) = 1, m ≤ n Ψm,n(s, t) = √m + n

• 1

√n cos s ei√ n

m t,

1 √m sin s ei√ m

n t

Clifford torus T1,1 ≡ Ψ1,1(s, t) = √ 2eit(cos s, sin s ) only embedded [Nemirovski, 2009] A Klein bottle does not admit a Lagrangian embedding in C2

TRANSLATING SOLITONS

See Ana M. Lerma’s poster

Translating solitons for Lagrangian mean curvature flow in complex Euclidean plane

• Abstract. Using certain solutions of the curve shortening flow, including self-shrinking and self-expanding curves or spirals, we construct and characterize many new examples of translating solitons for mean curvature

• a < 0, self-shrinker

• R2n ≡ Cn, , , J
• complex Euclidean space

• ∂

• t=0 Hamiltonian: V = J∇f, f ∈ C∞

• φ∗ , =
• |α|2 + |ω|2

• a + κ2

• + b κ′α = 0

• ⊲ ϕ = 0 :

• α0(t) = t, ωE self-expanding curves (Joyce, Lee and Tsui examples [3])

• ◮ ϕ = π/2 :

• HSL TRANSLATING SOLITONS

• s2−t2

• ,

• s2

• ,

• Theorem. φ : M2 → C2 Lagrangian H = e⊥

• Corollary. φ : M → C2 HSL (non totally geodesic)

References

Castro, Ildefonso; Lerma, Ana M. Hamiltonian stationary self-similar solutions for Lagrangian mean curvature flow in complex Euclidean plane, Proc. Amer. Math. Soc. 138 (2010), 1821–1832. Castro, Ildefonso; Lerma, Ana M. Translating solitons for Lagrangian mean curvature flow in complex Euclidean plane, Int. J. Math. 23 (2012). Castro, Ildefonso; Lerma, Ana M. The Clifford torus as a self-shrinker for the Lagrangian mean curvature flow, arXiv:1202.2555 [math.DG], submitted.