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

Soliton solutions for Lagrangian MCF ⊠ Self-similar solutions for Lagrangian MCF: H = ± φ ⊥ Examples: self-shrinkers product of circles and lines S 1 × R ⊂ C 2 , S 1 × S 1 ⊂ C 2

Soliton solutions for Lagrangian MCF ⊠ Self-similar solutions for Lagrangian MCF: H = ± φ ⊥ Examples: self-shrinkers product of circles and lines S 1 × R ⊂ C 2 , S 1 × S 1 ⊂ C 2 � [Anciaux, 2006] Construction of Lagrangian self-similar solutions to the mean curvature flow in C n . 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 S 1 × R ⊂ C 2 , S 1 × S 1 ⊂ C 2 � [Anciaux, 2006] Construction of Lagrangian self-similar solutions to the mean curvature flow in C n . 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 S 1 × R ⊂ C 2 , S 1 × S 1 ⊂ C 2 � [Anciaux, 2006] Construction of Lagrangian self-similar solutions to the mean curvature flow in C n . 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 S 1 × R ⊂ C 2 , S 1 × S 1 ⊂ C 2 � [Anciaux, 2006] Construction of Lagrangian self-similar solutions to the mean curvature flow in C n . 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, J e � + constant

Soliton solutions for Lagrangian MCF ⊠ Translating solitons for Lagrangian MCF: H = e ⊥ β = −� φ, J e � + 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, J e � + 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, J e � + 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, J e � + 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 Φ δ : R 2 → C 2 , δ > 0

Self-expanders Φ δ : R 2 → C 2 , δ > 0 � i s δ cosh t e − i s c δ , t δ sinh t e i c δ s � Φ δ ( s , t ) = HSL H δ = Φ ⊥ s δ = sinh δ , c δ = cosh δ , t δ = tanh δ δ

Self-expanders Φ δ : R 2 → C 2 , δ > 0 � i s δ cosh t e − i s c δ , t δ sinh t e i c δ s � Φ δ ( s , t ) = HSL H δ = Φ ⊥ s δ = sinh δ , c δ = cosh δ , t δ = tanh δ δ cosh 2 δ / ∈ Q , Φ δ embedded plane;

Self-expanders Φ δ : R 2 → C 2 , δ > 0 � i s δ cosh t e − i s c δ , t δ sinh t e i c δ s � Φ δ ( s , t ) = HSL H δ = Φ ⊥ s δ = sinh δ , c δ = cosh δ , t δ = tanh δ δ cosh 2 δ / ∈ Q , Φ δ embedded plane; cosh 2 δ = p / q ∈ Q , ( p , q ) = 1 Φ p , q : R 2 → C 2 , p > q � [Lee & Wang, 2009] � i √ q cosh t e − i √ q √ p sinh t e i √ p Φ p , q ( s , t ) = √ p − q � p s , 1 q s ◮ Φ p , q ( s + 2 π √ pq , t ) = Φ p , q ( s , t ), ∀ ( s , t ) ∈ R 2 cylinder

Self-expanders Φ δ : R 2 → C 2 , δ > 0 � i s δ cosh t e − i s c δ , t δ sinh t e i c δ s � Φ δ ( s , t ) = HSL H δ = Φ ⊥ s δ = sinh δ , c δ = cosh δ , t δ = tanh δ δ cosh 2 δ / ∈ Q , Φ δ embedded plane; cosh 2 δ = p / q ∈ Q , ( p , q ) = 1 Φ p , q : R 2 → C 2 , p > q � [Lee & Wang, 2009] � i √ q cosh t e − i √ q √ p sinh t e i √ p Φ p , q ( s , t ) = √ p − q � p s , 1 q s ◮ Φ p , q ( s + 2 π √ pq , t ) = Φ p , q ( s , t ), ∀ ( s , t ) ∈ R 2 cylinder ◮ p odd, q even: Φ p , q ( s + π √ pq , − t ) = Φ p , q ( s , t ), ∀ ( s , t ) ∈ R 2 Moebius strip

Plane Φ δ , cosh 2 δ / ∈ Q

Cylinder Φ 3 , 1

Moebius strip Φ 3 , 2

Self-shrinkers Υ γ : R 2 → C 2 , 0 < γ < π/ 2

Self-shrinkers Υ γ : R 2 → C 2 , 0 < γ < π/ 2 � c γ , t γ sinh t e − i c γ s � i s Υ γ ( s , t ) = − i s γ cosh t e HSL H γ = − Υ ⊥ s γ = sin γ , c γ = cos γ , t γ = tan γ γ

Self-shrinkers Υ γ : R 2 → C 2 , 0 < γ < π/ 2 � c γ , t γ sinh t e − i c γ s � i s Υ γ ( s , t ) = − i s γ cosh t e HSL H γ = − Υ ⊥ s γ = sin γ , c γ = cos γ , t γ = tan γ γ cos 2 γ / ∈ Q , Υ γ embedded plane;

Self-shrinkers Υ γ : R 2 → C 2 , 0 < γ < π/ 2 � c γ , t γ sinh t e − i c γ s � i s Υ γ ( s , t ) = − i s γ cosh t e HSL H γ = − Υ ⊥ s γ = sin γ , c γ = cos γ , t γ = tan γ γ cos 2 γ / ∈ Q , Υ γ embedded plane; cos 2 γ = p / q ∈ Q , ( p , q ) = 1 Υ p , q : R 2 → C 2 , p < q � [Lee & Wang, 2009] � − i √ q cosh t e i √ q √ p sinh t e − i √ p Υ p , q ( s , t ) = √ q − p p s , 1 � q s ◮ Υ p , q ( s + 2 π √ pq , t ) = Υ p , q ( s , t ), ∀ ( s , t ) ∈ R 2 cylinder

Self-shrinkers Υ γ : R 2 → C 2 , 0 < γ < π/ 2 � c γ , t γ sinh t e − i c γ s � i s Υ γ ( s , t ) = − i s γ cosh t e HSL H γ = − Υ ⊥ s γ = sin γ , c γ = cos γ , t γ = tan γ γ cos 2 γ / ∈ Q , Υ γ embedded plane; cos 2 γ = p / q ∈ Q , ( p , q ) = 1 Υ p , q : R 2 → C 2 , p < q � [Lee & Wang, 2009] � − i √ q cosh t e i √ q √ p sinh t e − i √ p Υ p , q ( s , t ) = √ q − p p s , 1 � q s ◮ Υ p , q ( s + 2 π √ pq , t ) = Υ p , q ( s , t ), ∀ ( s , t ) ∈ R 2 cylinder ◮ q even, p odd: Υ p , q ( s + π √ pq , − t ) = Υ p , q ( s , t ), ∀ ( s , t ) ∈ R 2 Moebius strip

Plane Υ γ , cos 2 γ / ∈ Q

Cylinder Υ 1 , 3

Moebius strip Υ 1 , 2

Self-shrinkers Ψ ν : S 1 × R → C 2 , ν > 0

Self-shrinkers Ψ ν : S 1 × R → C 2 , ν > 0 � s ν , t ν sin s e i s ν t � i t Ψ ν ( e i s , t ) = c ν cos s e HSL H ν = − Ψ ⊥ s ν = sinh ν , c ν = cosh ν , t ν = coth ν ν

Self-shrinkers Ψ ν : S 1 × R → C 2 , ν > 0 � s ν , t ν sin s e i s ν t � i t Ψ ν ( e i s , t ) = c ν cos s e HSL H ν = − Ψ ⊥ s ν = sinh ν , c ν = cosh ν , t ν = coth ν ν sinh 2 ν / ∈ Q , Ψ ν embedded cylinder;

Self-shrinkers Ψ ν : S 1 × R → C 2 , ν > 0 � s ν , t ν sin s e i s ν t � i t Ψ ν ( e i s , t ) = c ν cos s e HSL H ν = − Ψ ⊥ s ν = sinh ν , c ν = cosh ν , t ν = coth ν ν sinh 2 ν / ∈ Q , Ψ ν embedded cylinder; sinh 2 ν = m / n ∈ Q , ( m , n )=1 Ψ m , n : S 1 × R → C 2 , ( m , n ) = 1 � [Lee & Wang, 2010] � 1 √ n cos s e i √ n √ m sin s e i √ m √ 1 � m t , n t Ψ m , n ( s , t ) = m + n ◮ Ψ m , n ( s +2 π, t )=Ψ m , n ( s , t )=Ψ m , n ( s , t +2 π √ mn ), ∀ ( s , t ) ∈ R 2

Family Ψ m , n

Family Ψ m , n ◮ m , n odd: Ψ m , n ( s + π, t + π √ mn ) = Ψ m , n ( s , t ), ∀ ( s , t ) ∈ R 2 T m , n = Ψ m , n ( R 2 / Λ m , n ) torus ◮ m odd, n even: Ψ m , n (2 π − s , t + π √ mn )=Ψ m , n ( s , t ), ∀ ( s , t ) ∈ R 2 Klein bottle ◮ m even, n odd: Ψ m , n ( π − s , t + π √ mn )=Ψ m , n ( s , t ), ∀ ( s , t ) ∈ R 2 Klein bottle

Family Ψ m , n ◮ m , n odd: Ψ m , n ( s + π, t + π √ mn ) = Ψ m , n ( s , t ), ∀ ( s , t ) ∈ R 2 T m , n = Ψ m , n ( R 2 / Λ m , n ) torus ◮ m odd, n even: Ψ m , n (2 π − s , t + π √ mn )=Ψ m , n ( s , t ), ∀ ( s , t ) ∈ R 2 Klein bottle ◮ m even, n odd: Ψ m , n ( π − s , t + π √ mn )=Ψ m , n ( s , t ), ∀ ( s , t ) ∈ R 2 Klein bottle Clifford torus T 1 , 1 only one embedded

Family Ψ m , n ◮ m , n odd: Ψ m , n ( s + π, t + π √ mn ) = Ψ m , n ( s , t ), ∀ ( s , t ) ∈ R 2 T m , n = Ψ m , n ( R 2 / Λ m , n ) torus ◮ m odd, n even: Ψ m , n (2 π − s , t + π √ mn )=Ψ m , n ( s , t ), ∀ ( s , t ) ∈ R 2 Klein bottle ◮ m even, n odd: Ψ m , n ( π − s , t + π √ mn )=Ψ m , n ( s , t ), ∀ ( s , t ) ∈ R 2 Klein bottle Clifford torus T 1 , 1 only one embedded 4( m + n ) 2 π 2  , m or n even √ mn   Willmore ( T m , n ) = 2 Area ( T m , n ) = 2( m + n ) 2 π 2  , m and n odd  √ mn

Clifford torus

Torus Ψ 1 , 3

Klein bottle Ψ 1 , 2

Classification results

Classification results Theorem φ : M 2 → C 2 HSL self-similar solution for MCF (a) φ self-expander (H = φ ⊥ ) ∼ Φ δ : R 2 → C 2 , δ > 0 loc ⇒ φ loc (b) φ self-shrinker (H = − φ ⊥ ) ⇒ φ ∼ (i) S 1 × R (ii) S 1 × S 1 (iii) Υ γ : R 2 → C 2 , 0 < γ < π/ 2 (iv) Ψ ν : S 1 × R → C 2 , ν > 0

Classification results Theorem φ : M 2 → C 2 HSL self-similar solution for MCF (a) φ self-expander (H = φ ⊥ ) ∼ Φ δ : R 2 → C 2 , δ > 0 loc ⇒ φ loc (b) φ self-shrinker (H = − φ ⊥ ) ⇒ φ ∼ (i) S 1 × R (ii) S 1 × S 1 (iii) Υ γ : R 2 → C 2 , 0 < γ < π/ 2 (iv) Ψ ν : S 1 × R → C 2 , ν > 0 Corollary φ : M → C 2 HSL self-similar solution for MCF M compact orientable ⇒ φ ( M ) ∼ T m , n

THE CLIFFORD TORUS AS A SELF-SHRINKER

Self-shrinkers: notion and examples

Self-shrinkers: notion and examples φ : M n → R m self-shrinker if H = − φ ⊥ ( H = trace σ )

Self-shrinkers: notion and examples φ : M n → R m self-shrinker if H = − φ ⊥ ( H = trace σ ) ◮ F ( p , t ) = √ 1 − 2 t Φ( p ), 0 ≤ t < 1 / 2, solution to (MCF)

Self-shrinkers: notion and examples φ : M n → R m self-shrinker if H = − φ ⊥ ( H = trace σ ) ◮ F ( p , t ) = √ 1 − 2 t Φ( p ), 0 ≤ t < 1 / 2, solution to (MCF) Example (Sphere) S n ( √ n ) ֒ → R n +1 , | σ | 2 ≡ 1

Self-shrinkers: notion and examples φ : M n → R m self-shrinker if H = − φ ⊥ ( H = trace σ ) ◮ F ( p , t ) = √ 1 − 2 t Φ( p ), 0 ≤ t < 1 / 2, solution to (MCF) Example (Sphere) S n ( √ n ) ֒ → R n +1 , | σ | 2 ≡ 1 Example (Clifford) S n 1 ( √ n 1 ) × S n 2 ( √ n 2 ) ֒ → R n +2 , n 1 , n 2 ∈ N , n 1 + n 2 = n , | σ | 2 ≡ 2

Self-shrinkers: notion and examples φ : M n → R m self-shrinker if H = − φ ⊥ ( H = trace σ ) ◮ F ( p , t ) = √ 1 − 2 t Φ( p ), 0 ≤ t < 1 / 2, solution to (MCF) Example (Sphere) S n ( √ n ) ֒ → R n +1 , | σ | 2 ≡ 1 Example (Clifford) S n 1 ( √ n 1 ) × S n 2 ( √ n 2 ) ֒ → R n +2 , n 1 , n 2 ∈ N , n 1 + n 2 = n , | σ | 2 ≡ 2 Example (Product of n -circles) n ) · · · × S 1 ֒ → R 2 n , | σ | 2 ≡ n , Lagrangian in R 2 n ≡ C n S 1 ×

Self-shrinkers: notion and examples φ : M n → R m self-shrinker if H = − φ ⊥ ( H = trace σ ) ◮ F ( p , t ) = √ 1 − 2 t Φ( p ), 0 ≤ t < 1 / 2, solution to (MCF) Example (Sphere) S n ( √ n ) ֒ → R n +1 , | σ | 2 ≡ 1 Example (Clifford) S n 1 ( √ n 1 ) × S n 2 ( √ n 2 ) ֒ → R n +2 , n 1 , n 2 ∈ N , n 1 + n 2 = n , | σ | 2 ≡ 2 Example (Product of n -circles) n ) · · · × S 1 ֒ → R 2 n , | σ | 2 ≡ n , Lagrangian in R 2 n ≡ C n S 1 × Example (Product of a circle and ( n − 1)-sphere) ( e it , ( x 1 , . . . , x n )) �→ √ n e it ( x 1 , . . . , x n ) S 1 × S n − 1 → C n ≡ R 2 n , | σ | 2 ≡ (3 n − 2) / n , Lagrangian in C n ≡ R 2 n

Case n = 1: Self-shrinking curves

Case n = 1: Self-shrinking curves − → κ α = − α ⊥

Case n = 1: Self-shrinking curves − → κ α = − 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 : M n → R n +1 , M compact self-shrinker H ≥ 0 ⇒ M n ≡ S n ( √ n )

Classification and rigidity results, n ≥ 2 � [Huisken, J. Diff. Geom. 1990] φ : M n → R n +1 , M compact self-shrinker H ≥ 0 ⇒ M n ≡ S n ( √ n ) � [Smoczyk, Int. Math. Res. Not. 2005] φ : M n → R m , M compact self-shrinker | H | > 0; ∇ ⊥ ν = 0, ν = H / | H | � M n spherical: M n → S m − 1 ( √ n ), ˆ H = 0

Classification and rigidity results, n ≥ 2 � [Huisken, J. Diff. Geom. 1990] φ : M n → R n +1 , M compact self-shrinker H ≥ 0 ⇒ M n ≡ S n ( √ n ) � [Smoczyk, Int. Math. Res. Not. 2005] φ : M n → R m , M compact self-shrinker | H | > 0; ∇ ⊥ ν = 0, ν = H / | H | � M n spherical: M n → S m − 1 ( √ n ), ˆ H = 0 � [Le & Sesum, Comm. Anal. Geom. 2011] � [Cao & Li, Calc. Var. PDE 2012] φ : M n → R m , M compact self-shrinker | σ | 2 ≤ 1 ⇒ | σ | 2 ≡ 1, M n ≡ S n ( √ n ) ⊂ R n +1

Classification and rigidity results, n ≥ 2 � [Li & Wei, preprint 2012] φ : M n → R n +2 , M compact self-shrinker

Classification and rigidity results, n ≥ 2 � [Li & Wei, preprint 2012] φ : M n → R n +2 , M compact self-shrinker 1 ≤ | σ | 2 ≤ 2 | H | > 0; ∇ ⊥ ν = 0, ν = H / | H |

Classification and rigidity results, n ≥ 2 � [Li & Wei, preprint 2012] φ : M n → R n +2 , M compact self-shrinker 1 ≤ | σ | 2 ≤ 2 | H | > 0; ∇ ⊥ ν = 0, ν = H / | H | 1 either | σ | 2 ≡ 1, M ≡ S n ( √ n ) ⊂ R n +1 2 or | σ | 2 ≡ 2, M ≡ S n 1 ( √ n 1 ) × S n 2 ( √ n 2 ) ⊂ R n +2 , n 1 + n 2 = n

Classification and rigidity results, n ≥ 2 � [Li & Wei, preprint 2012] φ : M n → R n +2 , M compact self-shrinker 1 ≤ | σ | 2 ≤ 2 | H | > 0; ∇ ⊥ ν = 0, ν = H / | H | 1 either | σ | 2 ≡ 1, M ≡ S n ( √ n ) ⊂ R n +1 2 or | σ | 2 ≡ 2, M ≡ S n 1 ( √ n 1 ) × S n 2 ( √ n 2 ) ⊂ R n +2 , n 1 + n 2 = n φ : M 2 → R 2+ p , M compact self-shrinker 1 ≤ | σ | 2 ≤ 5 | H | > 0; ∇ ⊥ ν = 0, ν = H / | H | 3

Classification and rigidity results, n ≥ 2 � [Li & Wei, preprint 2012] φ : M n → R n +2 , M compact self-shrinker 1 ≤ | σ | 2 ≤ 2 | H | > 0; ∇ ⊥ ν = 0, ν = H / | H | 1 either | σ | 2 ≡ 1, M ≡ S n ( √ n ) ⊂ R n +1 2 or | σ | 2 ≡ 2, M ≡ S n 1 ( √ n 1 ) × S n 2 ( √ n 2 ) ⊂ R n +2 , n 1 + n 2 = n φ : M 2 → R 2+ p , M compact self-shrinker 1 ≤ | σ | 2 ≤ 5 | H | > 0; ∇ ⊥ ν = 0, ν = H / | H | 3 √ 1 either | σ | 2 ≡ 1, M ≡ S 2 ( 2) ⊂ R 3 √ √ 2 or | σ | 2 ≡ 5 / 3, M ≡ S 2 ( → R 5 Veronese 6) → S 4 ( 2) ֒

Classification and rigidity results, n ≥ 2 � [Li & Wei, preprint 2012] φ : M n → R n +2 , M compact self-shrinker 1 ≤ | σ | 2 ≤ 2 | H | > 0; ∇ ⊥ ν = 0, ν = H / | H | 1 either | σ | 2 ≡ 1, M ≡ S n ( √ n ) ⊂ R n +1 2 or | σ | 2 ≡ 2, M ≡ S n 1 ( √ n 1 ) × S n 2 ( √ n 2 ) ⊂ R n +2 , n 1 + n 2 = n φ : M 2 → R 2+ p , M compact self-shrinker 1 ≤ | σ | 2 ≤ 5 | H | > 0; ∇ ⊥ ν = 0, ν = H / | H | 3 √ 1 either | σ | 2 ≡ 1, M ≡ S 2 ( 2) ⊂ R 3 √ √ 2 or | σ | 2 ≡ 5 / 3, M ≡ S 2 ( → R 5 Veronese 6) → S 4 ( 2) ֒ 3 ≤ | σ | 2 ≤ 11 5 | H | > 0; ∇ ⊥ ν = 0, ν = H / | H | 6

Classification and rigidity results, n ≥ 2 � [Li & Wei, preprint 2012] φ : M n → R n +2 , M compact self-shrinker 1 ≤ | σ | 2 ≤ 2 | H | > 0; ∇ ⊥ ν = 0, ν = H / | H | 1 either | σ | 2 ≡ 1, M ≡ S n ( √ n ) ⊂ R n +1 2 or | σ | 2 ≡ 2, M ≡ S n 1 ( √ n 1 ) × S n 2 ( √ n 2 ) ⊂ R n +2 , n 1 + n 2 = n φ : M 2 → R 2+ p , M compact self-shrinker 1 ≤ | σ | 2 ≤ 5 | H | > 0; ∇ ⊥ ν = 0, ν = H / | H | 3 √ 1 either | σ | 2 ≡ 1, M ≡ S 2 ( 2) ⊂ R 3 √ √ 2 or | σ | 2 ≡ 5 / 3, M ≡ S 2 ( → R 5 Veronese 6) → S 4 ( 2) ֒ 3 ≤ | σ | 2 ≤ 11 5 | H | > 0; ∇ ⊥ ν = 0, ν = H / | H | 6 √ √ 1 either | σ | 2 ≡ 5 / 3, M ≡ S 2 ( → R 5 Veronese 6) → S 4 ( 2) ֒ √ √ 2 or | σ | 2 ≡ 11 / 6, M ≡ S 2 ( → R 7 Boruvka 12) → S 6 ( 2) ֒

Our contribution (arbitrary dimension and codimension) Theorem A φ : M n → R n + p , M compact self-shrinker | H | 2 constant or | H | 2 ≤ n or | H | 2 ≥ n | σ | 2 ≤ 3 p − 4 2 p − 3

Our contribution (arbitrary dimension and codimension) Theorem A φ : M n → R n + p , M compact self-shrinker | H | 2 constant or | H | 2 ≤ n or | H | 2 ≥ n | σ | 2 ≤ 3 p − 4 2 p − 3 Then: 1 either | σ | 2 ≡ 1 , M ≡ S n ( √ n ) ⊂ R n +1 [p = 1 ]

Our contribution (arbitrary dimension and codimension) Theorem A φ : M n → R n + p , M compact self-shrinker | H | 2 constant or | H | 2 ≤ n or | H | 2 ≥ n | σ | 2 ≤ 3 p − 4 2 p − 3 Then: 1 either | σ | 2 ≡ 1 , M ≡ S n ( √ n ) ⊂ R n +1 [p = 1 ] 2 or | σ | 2 ≡ 3 p − 4 2 p − 3 , (i) either M ≡ S n 1 ( √ n 1 ) × S n 2 ( √ n 2 ) ⊂ R n +2 , n 1 + n 2 = n, (with | σ | 2 ≡ 2 ) [p = 2 ] √ 6) ⊂ R 5 (with | σ | 2 ≡ 5 / 3 ) [n = 2 , p = 3 ] (ii) or Veronese M ≡ S 2 (

Proof of Theorem A

Proof of Theorem A H = − φ ⊥ ⇒ △| φ | 2 = 2( n − | H | 2 )