A Quantitative Look at Lagrangian Cobordisms

Transcription

1 A Quantitative Look at Lagrangian Cobordisms Lisa Traynor Bryn Mawr College Joint work with Joshua M. Sabloff, Haverford College December 2016 Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 1 / 36

2 Lagrangians and Legendrians y z y x x Symplectic Manifold (X 2n, ω) Contact Manifold (Y 2n+1, ξ) Lagrangian Submanifold L n : ω TL 0 Legendrian Submanifold Λ n : T Λ ξ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 2 / 36

3 Lagrangians and Legendrians y z y x x Symplectic Manifold (X 2n, ω) Exact Symplectic : ω = dλ Lagrangian Submanifold L n : ω TL 0 Exact Lagrangian: λ = df Contact Manifold (Y 2n+1, ξ) Legendrian Submanifold Λ n : T Λ ξ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 2 / 36

4 The Symplectization of a Contact Manifold Standard Contact Manifold: ( R 2n+1, ker α ) J 1 (R n ) = T R n R = R 2n+1, α = dz i y idx i Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 3 / 36

5 The Symplectization of a Contact Manifold Standard Contact Manifold: ( R 2n+1, ker α ) J 1 (R n ) = T R n R = R 2n+1, α = dz i y idx i Symplectization: ( R R 2n+1, d(e s α) ) R 2n+1 s Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 3 / 36

6 The Symplectization of a Contact Manifold Standard Contact Manifold: ( R 2n+1, ker α ) J 1 (R n ) = T R n R = R 2n+1, α = dz i y idx i Symplectization: ( R R 2n+1, d(e s α) ) R 2n+1 s There are no closed, exact Lagrangians (Gromov); Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 3 / 36

7 The Symplectization of a Contact Manifold Standard Contact Manifold: ( R 2n+1, ker α ) J 1 (R n ) = T R n R = R 2n+1, α = dz i y idx i Symplectization: ( R R 2n+1, d(e s α) ) R 2n+1 s There are no closed, exact Lagrangians (Gromov); For a Legendrian Λ, the cylinder R Λ is an exact Lagrangian. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 3 / 36

8 Lagrangian Cobordisms between Legendrians A Lagrangian cobordism from Λ to Λ + means: s + + R 2n+2 s s Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 4 / 36

9 Lagrangian Cobordisms between Legendrians A Lagrangian cobordism from Λ to Λ + means: s + + R 2n+2 s s Λ ± are Legendrian submanifolds in {s = s ± }; Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 4 / 36

10 Lagrangian Cobordisms between Legendrians A Lagrangian cobordism from Λ to Λ + means: s + + R 2n+2 s s Λ ± are Legendrian submanifolds in {s = s ± }; L is Lagrangian and cylindrical over Λ ± at ± : L = R Λ ± outside [s, s +]; Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 4 / 36

11 Lagrangian Cobordisms between Legendrians A Lagrangian cobordism from Λ to Λ + means: s + + R 2n+2 s s Λ ± are Legendrian submanifolds in {s = s ± }; L is Lagrangian and cylindrical over Λ ± at ± : L is embedded and exact: L = R Λ ± outside [s, s +]; e s α L = df, f = constant ± outside [s, s +]. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 4 / 36

12 Lagrangian Cobordisms between Legendrians A Lagrangian cobordism from Λ to Λ + means: s + + R 2n+2 s s Λ ± are Legendrian submanifolds in {s = s ± }; L is Lagrangian and cylindrical over Λ ± at ± : L is embedded and exact: L = R Λ ± outside [s, s +]; e s α L = df, f = constant ± outside [s, s +]. Arise in relative SFT (Eliashberg-Givental-Hofer) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 4 / 36

13 Qualitative Questions Given Λ, Λ + R 2n+1, does there exist a Lagrangian cobordism between them? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 5 / 36

14 Qualitative Questions E E Given Λ, Λ + R 2n+1, does there exist a Lagrangian cobordism between them? Non-symmetric relation! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 5 / 36

15 Qualitative Questions E E Given Λ, Λ + R 2n+1, does there exist a Lagrangian cobordism between them? Non-symmetric relation! How topologically rigid are Lagrangian cobordisms? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 5 / 36

16 Qualitative Questions E E Given Λ, Λ + R 2n+1, does there exist a Lagrangian cobordism between them? Non-symmetric relation! How topologically rigid are Lagrangian cobordisms? Fillings realize 4-ball genus! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 5 / 36

17 Qualitative Questions E E Given Λ, Λ + R 2n+1, does there exist a Lagrangian cobordism between them? Non-symmetric relation! How topologically rigid are Lagrangian cobordisms? Fillings realize 4-ball genus! A variety of qualitative questions have been studied by: Chantraine, Ekholm, Honda, Kálmán, Dimitroglou Rizell, Ghiggini, Golovko, Cornwell, Ng, Sivek, Bourgeois, Sabloff, Traynor, Capovilla-Searle, Hayden, Pan,... Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 5 / 36

18 Quantitative Questions (Length) Given Λ, Λ + R 2n+1, what is the minimal length" of any cobordism between them? h = s + 0 =s_ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 6 / 36

19 Quantitative Questions (Length) Given Λ, Λ + R 2n+1, what is the minimal length" of any cobordism between them? h = s + 0 =s_ (Width) Given a Lagrangian cobordism, what is its width"? B(c) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 6 / 36

20 Outline 1 Constructions of Lagrangian Cobordisms

21 Outline 1 Constructions of Lagrangian Cobordisms 2 Length of a Lagrangian cobordism

22 Outline 1 Constructions of Lagrangian Cobordisms 2 Length of a Lagrangian cobordism 3 Width of a Lagrangian Cobordism

23 Outline 1 Constructions of Lagrangian Cobordisms 2 Length of a Lagrangian cobordism 3 Width of a Lagrangian Cobordism Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 8 / 36

24 E Constructions of Lagrangian Concordances Isotopy Lemma (Eliashberg, Chantraine, Golovko, Ekholm-Honda-Kálmán,... ) Suppose Λ and Λ + are Legendrian isotopic. Then there exists a Lagrangian cobordism from Λ to Λ +. + R 3 s _ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 9 / 36

25 E Constructions of Lagrangian Concordances Isotopy Lemma (Eliashberg, Chantraine, Golovko, Ekholm-Honda-Kálmán,... ) Suppose Λ and Λ + are Legendrian isotopic. Then there exists a Lagrangian cobordism from Λ to Λ +. + R 3 s _ Remark: The Lagrangian is not the trace of the isotopy. Most slices of the Lagrangian will not be Legendrian. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 9 / 36

26 Lagrangian Concordances from Isotopy E E Qualitatively Symmetric Concordances: _ 1 _ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 10 / 36

27 Constructions of Lagrangian Cobordisms Theorem (Dimitroglou Rizell, Ekholm-Honda-Kálmán, Bourgeois-Sabloff-T ) If Λ + is obtained from Λ by a cusp-surgery", Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 11 / 36

28 Constructions of Lagrangian Cobordisms Theorem (Dimitroglou Rizell, Ekholm-Honda-Kálmán, Bourgeois-Sabloff-T ) If Λ + is obtained from Λ by a cusp-surgery", Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 11 / 36

29 Constructions of Lagrangian Cobordisms Theorem (Dimitroglou Rizell, Ekholm-Honda-Kálmán, Bourgeois-Sabloff-T ) If Λ + is obtained from Λ by a cusp-surgery", then there exists a Lagrangian cobordism from Λ to Λ +. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 11 / 36

30 Construction Example Lagrangian genus 1 filling of a Legendrian m(5 2 ): Legendrian isotopy and cusp pinches as you move up! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 12 / 36

31 Outline 1 Constructions of Lagrangian Cobordisms 2 Length of a Lagrangian cobordism 3 Width of a Lagrangian Cobordism Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 13 / 36

32 Length Question: Given Λ, Λ + R 2n+1, what is the minimal length" of any cobordism between them? h = s + 0 =s_ minimal length = inf{h : Lagrangian cobordism from Λ to Λ + that is cylindrical outside[0, h]}. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 14 / 36

33 Flexibility Theorem (Sabloff-T, 16: Selecta Mathematica) There exists an arbitrarily short Lagrangian cobordism between 1 a Legendrian and its vertical translate, h~0 + s 0 _ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 15 / 36

34 Flexibility Theorem (Sabloff-T, 16: Selecta Mathematica) There exists an arbitrarily short Lagrangian cobordism between 1 a Legendrian and its vertical translate, h~0 + s 0 _ 2 a Legendrian and its horizontal translate, h~0 s 0 Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 15 / 36

35 Flexibility Theorem (Sabloff-T, 16: Selecta Mathematica) There exists an arbitrarily short Lagrangian cobordism between 1 a Legendrian and its vertical translate, h~0 + s 0 _ 2 a Legendrian and its horizontal translate, h~0 s 0 3 a Legendrian and its vertical expansion. h~0 + s 0 _ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 15 / 36

36 Rigidity Theorem (Sabloff-T, 16) There exist obstructions to arbitrarily short Lagrangian cobordisms between 1 a Legendrian and its vertical contraction; h~ln s 0 2 _ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 16 / 36

37 Rigidity Theorem (Sabloff-T, 16) There exist obstructions to arbitrarily short Lagrangian cobordisms between 1 a Legendrian and its vertical contraction; h~ln s 0 2 _ 2 vertically shifted Hopf links: h 1 v s 0 1 u h { ( ) ln 1 u 1 v, if u v, ln ( ) u v, if u v. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 16 / 36

38 Lower Bound to Length (Step 1) Assign capacities" to a Legendrian c(λ, ε, θ) R >0 { }, ε is an augmentation of the DGA A(Λ), ε : (A(Λ), ) (F 2, 0), θ LCH (Λ, ε). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 17 / 36

39 Lower Bound to Length (Step 1) Assign capacities" to a Legendrian c(λ, ε, θ) R >0 { }, ε is an augmentation of the DGA A(Λ), ε : (A(Λ), ) (F 2, 0), θ LCH (Λ, ε). Example: r 0 λ LCH 1 (U(r), ε); c(u(r), ε, λ) = r. Fundamental Class Fundamental Capacity Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 17 / 36

40 Lower Bound to Length (Step 1) Assign capacities" to a Legendrian c(λ, ε, θ) R >0 { }, ε is an augmentation of the DGA A(Λ), ε : (A(Λ), ) (F 2, 0), θ LCH (Λ, ε). Example: r 0 λ LCH 1 (U(r), ε); c(u(r), ε, λ) = r. Fundamental Class Fundamental Capacity For θ 0, c(λ, ε, θ) is always the height of a Reeb chord! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 17 / 36

41 Lower Bound to Length (Step 2) From ε, θ for Λ and Lagrangian cobordism L from Λ to Λ +, get induced ε +, θ + for Λ +. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 18 / 36

42 Lower Bound to Length (Step 2) From ε, θ for Λ and Lagrangian cobordism L from Λ to Λ +, get induced ε +, θ + for Λ +. [Ekholm-Honda-Kálmán] Λ + A(Λ + ) Φ(L) ε + LCH (Λ +, ε + ) θ + Ψ L,ε Λ A(Λ ) ε F 2 LCH (Λ, ε ) θ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 18 / 36

43 Lower Bound to Length (Step 2) From ε, θ for Λ and Lagrangian cobordism L from Λ to Λ +, get induced ε +, θ + for Λ +. [Ekholm-Honda-Kálmán] Λ + A(Λ + ) Φ(L) ε + LCH (Λ +, ε + ) θ + Ψ L,ε Λ A(Λ ) ε F 2 LCH (Λ, ε ) θ Question: How do capacities c(λ +, ε +, θ + ) and c(λ, ε, θ ) compare? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 18 / 36

44 Lower Bound to Length (Step 3) Relate capacities for ends of a Lagrangian cobordism. Length-Capacity Inequality (Sabloff-T) If L is a Lagrangian cobordism from Λ to Λ + that is cylindrical outside [0, h], then e 0 c(λ, ε, θ ) e h c(λ +, ε +, θ + ). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 19 / 36

45 Lower Bound to Length (Step 3) Relate capacities for ends of a Lagrangian cobordism. Length-Capacity Inequality (Sabloff-T) If L is a Lagrangian cobordism from Λ to Λ + that is cylindrical outside [0, h], then e 0 c(λ, ε, θ ) e h c(λ +, ε +, θ + ). Remember ε +, θ + are induced by L. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 19 / 36

46 Lower Bound to Length (Step 3) Relate capacities for ends of a Lagrangian cobordism. Length-Capacity Inequality (Sabloff-T) If L is a Lagrangian cobordism from Λ to Λ + that is cylindrical outside [0, h], then e 0 c(λ, ε, θ ) e h c(λ +, ε +, θ + ). Remember ε +, θ + are induced by L. Get lower bounds to length of a cobordism! ( ) c(λ, ε, θ ) ln h. c(λ +, ε +, θ + ) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 19 / 36

47 Lower Bound to Length of a Contraction h 1 + s 0 2 _ By Length-Capacity Inequality: ( ) ( ) 2 c(u(2)), ε, λ ) ln = ln 1 c(u(1)), ε +, λ + ) = ln ( ) c(λ, ε, λ ) h. c(λ +, ε +, λ + ) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 20 / 36

48 Lower Bound to Length of a Contraction h 1 + s 0 2 _ By Length-Capacity Inequality: ( ) ( ) 2 c(u(2)), ε, λ ) ln = ln 1 c(u(1)), ε +, λ + ) = ln ( ) c(λ, ε, λ ) h. c(λ +, ε +, λ + ) Question: Can we get arbitrarily close to h = ln 2? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 20 / 36

49 Lower Bound to Length of a Contraction h 1 + s 0 2 _ By Length-Capacity Inequality: ( ) ( ) 2 c(u(2)), ε, λ ) ln = ln 1 c(u(1)), ε +, λ + ) = ln ( ) c(λ, ε, λ ) h. c(λ +, ε +, λ + ) Question: Can we get arbitrarily close to h = ln 2? Answer: Yes! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 20 / 36

50 Upper Bound to Length of a Contraction Lagrangian cobordism from Λ = U(2) to Λ + = U(1) of length A: Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 21 / 36

51 Upper Bound to Length of a Contraction Lagrangian cobordism from Λ = U(2) to Λ + = U(1) of length A: 1 1/2 Legendrian isotopy: λ s (t) = (x(t), ρ(s)y(t), ρ(s)z(t)) 0 A Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 21 / 36

52 Upper Bound to Length of a Contraction Lagrangian cobordism from Λ = U(2) to Λ + = U(1) of length A: 1 1/2 Legendrian isotopy: λ s (t) = (x(t), ρ(s)y(t), ρ(s)z(t)) 0 A Lagrangian immersion: Γ(s, t) = (s, x(t), ρ(s)y(t), ρ(s)z(t) + ρ (s)z(t)) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 21 / 36

53 Upper Bound to Length of a Contraction Lagrangian cobordism from Λ = U(2) to Λ + = U(1) of length A: 1 1/2 Legendrian isotopy: λ s (t) = (x(t), ρ(s)y(t), ρ(s)z(t)) 0 A Lagrangian immersion: Embedding condition: Γ(s, t) = (s, x(t), ρ(s)y(t), ρ(s)z(t) + ρ (s)z(t)) 1 e A /2 e s /2 e s d ds (es ρ(s)) 0 0 A Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 21 / 36

54 Upper Bound to Length of a Contraction Lagrangian cobordism from Λ = U(2) to Λ + = U(1) of length A: 1 1/2 Legendrian isotopy: λ s (t) = (x(t), ρ(s)y(t), ρ(s)z(t)) 0 A Lagrangian immersion: Embedding condition: So, Γ(s, t) = (s, x(t), ρ(s)y(t), ρ(s)z(t) + ρ (s)z(t)) 1 e A /2 e s /2 e s d ds (es ρ(s)) 0 0 A embedded Lagrangian cobordism when 1 < e A /2 ln 2 < A. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 21 / 36

55 Outline 1 Constructions of Lagrangian Cobordisms 2 Length of a Lagrangian cobordism 3 Width of a Lagrangian Cobordism Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 22 / 36

56 Width of a Symplectic Manifold B 2n (c) := { (x 1, y 1,..., x n, y n ) : π i (x 2 i + y 2 i ) c } (R 2n, ω 0 ). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 23 / 36

57 Width of a Symplectic Manifold B 2n (c) := { (x 1, y 1,..., x n, y n ) : π i (x 2 i + y 2 i ) c } (R 2n, ω 0 ). Width of a symplectic manifold (X, ω): w(x) := sup{c : ψ : B 2n (c) X, ψ ω = ω 0 }. B(c) X Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 23 / 36

58 Width of a Symplectic Manifold B 2n (c) := { (x 1, y 1,..., x n, y n ) : π i (x 2 i + y 2 i ) c } (R 2n, ω 0 ). Width of a symplectic manifold (X, ω): w(x) := sup{c : ψ : B 2n (c) X, ψ ω = ω 0 }. B(c) X We are working in X = R J 1 M: w(r J 1 M) =. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 23 / 36

59 Width of a Lagrangian Given a Lagrangian submanifold L (X, ω), relative width is: { w(x, L) = sup c ψ : B 2n (c) X, ψ ω = ω 0, ψ 1 (L) = B 2n (c) R n}. B(c) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 24 / 36

60 Width of a Lagrangian Given a Lagrangian submanifold L (X, ω), relative width is: { w(x, L) = sup c ψ : B 2n (c) X, ψ ω = ω 0, ψ 1 (L) = B 2n (c) R n}. B(c) Introduced by Barraud and Cornea, 05. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 24 / 36

61 Widths of Lagrangian Cobordisms Given a Lagrangian cobordism L, for a < b, L b a := {(s, x, y, z) L : a < s < b} (a, b) J 1 M. b B(c) a Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 25 / 36

62 Widths of Lagrangian Cobordisms Given a Lagrangian cobordism L, for a < b, L b a := {(s, x, y, z) L : a < s < b} (a, b) J 1 M. b B(c) a Question: Can we calculate w(l b a), for some L, a, b? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 25 / 36

63 Widths of Lagrangian Cobordisms Given a Lagrangian cobordism L, for a < b, L b a := {(s, x, y, z) L : a < s < b} (a, b) J 1 M. b B(c) a Question: Can we calculate w(l b a), for some L, a, b? Answer: Yes! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 25 / 36

64 Infinite Width Lemma For any Lagrangian cobordism L, for any a, w (L a ) =. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 26 / 36

65 Infinite Width Lemma For any Lagrangian cobordism L, for any a, w (L a ) =. Proof Sketch: B(r) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 26 / 36

66 Infinite Width Lemma For any Lagrangian cobordism L, for any a, w (L a ) =. Proof Sketch: B(R) B(r) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 26 / 36

67 Infinite Width Lemma For any Lagrangian cobordism L, for any a, w (L a ) =. Proof Sketch: B(R) B(r) Chop off top! We will consider: a =, s + b < +. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 26 / 36

68 Width of Cylindrical Lagrangian Cobordisms Cylindrical Lagrangian Cobordisms: L = R Λ, for a Legendrian Λ. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 27 / 36

69 Width of Cylindrical Lagrangian Cobordisms Cylindrical Lagrangian Cobordisms: L = R Λ, for a Legendrian Λ. Question: Can we calculate w((r Λ) b ) for some Λ and for some b? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 27 / 36

70 Width of Cylindrical Lagrangian Cobordisms Cylindrical Lagrangian Cobordisms: L = R Λ, for a Legendrian Λ. Question: Can we calculate w((r Λ) b ) for some Λ and for some b? Suffices to understand b = 0: Lemma For any Legendrian Λ, ( ) w (R Λ) b = e b w ( ) (R Λ) 0. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 27 / 36

71 Width of Cylindrical Lagrangian Cobordisms Cylindrical Lagrangian Cobordisms: L = R Λ, for a Legendrian Λ. Question: Can we calculate w((r Λ) b ) for some Λ and for some b? Suffices to understand b = 0: Lemma For any Legendrian Λ, ( ) w (R Λ) b = e b w ( ) (R Λ) 0. Question: Can we calculate w((r Λ) 0 ) for some Λ? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 27 / 36

72 Width of Cylindrical Lagrangian Cobordisms Cylindrical Lagrangian Cobordisms: L = R Λ, for a Legendrian Λ. Question: Can we calculate w((r Λ) b ) for some Λ and for some b? Suffices to understand b = 0: Lemma For any Legendrian Λ, ( ) w (R Λ) b = e b w ( ) (R Λ) 0. Question: Can we calculate w((r Λ) 0 ) for some Λ? Answer: Yes! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 27 / 36

73 Width of Cylinder over Legendrian Unknot r 0 r Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 28 / 36

74 Width of Cylinder over Legendrian Unknot r 0 r Theorem (Sabloff-T) w((r U(r)) 0 ) = 2r. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 28 / 36

75 Upperbound to Width of a Legendrian w ( (R U(r)) 0 ) 2r follows from: Theorem (Sabloff-T) Suppose Λ is a Legendrian that admits an augmentation. Then ( ) w (R Λ) 0 2c(Λ), where c(λ) is the minimum fundamental capacity (for any augmentation). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 29 / 36

76 Upperbound to Width of a Legendrian w ( (R U(r)) 0 ) 2r follows from: Theorem (Sabloff-T) Suppose Λ is a Legendrian that admits an augmentation. Then ( ) w (R Λ) 0 2c(Λ), where c(λ) is the minimum fundamental capacity (for any augmentation). size of ball 2 fundamental Reeb chord height" in = Λ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 29 / 36

77 Obstructions to Embeddings: Proof Sketch: Suppose there is an embedding ψ of B(α). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 30 / 36

78 Obstructions to Embeddings: Proof Sketch: Suppose there is an embedding ψ of B(α). By property of the fundamental class λ LCH (Λ, ε), through ψ(0) L there is a J-holomorphic disk" of area A c(λ, ε, λ). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 30 / 36

79 Obstructions to Embeddings: Proof Sketch: Suppose there is an embedding ψ of B(α). By property of the fundamental class λ LCH (Λ, ε), through ψ(0) L there is a J-holomorphic disk" of area A c(λ, ε, λ). There exists a holomorphic disk in B(α) with boundary in B(α) R n of area B A c(λ, ε, λ). By analytic continuation, this extends to a holomorphic disk with boundary in B(α) of area 2B 2c(Λ, ε, λ). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 30 / 36

80 Obstructions to Embeddings: Proof Sketch: Suppose there is an embedding ψ of B(α). By property of the fundamental class λ LCH (Λ, ε), through ψ(0) L there is a J-holomorphic disk" of area A c(λ, ε, λ). There exists a holomorphic disk in B(α) with boundary in B(α) R n of area B A c(λ, ε, λ). By analytic continuation, this extends to a holomorphic disk with boundary in B(α) of area 2B 2c(Λ, ε, λ). Classical Isoperimetric Inequality shows α 2B 2c(Λ, ε, λ). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 30 / 36

81 Lowerbound to width of a Legendrian 2r w ( (R U(r)) 0 ) follows from: Theorem (Sabloff-T) Suppose Λ has a vertically extendable" Reeb chord of height r. Then ( ) 2r w (R Λ) 0. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 31 / 36

82 Lowerbound to width of a Legendrian 2r w ( (R U(r)) 0 ) follows from: Theorem (Sabloff-T) Suppose Λ has a vertically extendable" Reeb chord of height r. Then ( ) 2r w (R Λ) 0. Vertically Extendable No Yes Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 31 / 36

83 Constructing Embeddings Proof Sketch: Ψ : R J 1 M T R + T M (s, x, y, z) ((e s, z), (x, e s y)) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 32 / 36

84 Constructing Embeddings Proof Sketch: Ψ : R J 1 M T R + T M (s, x, y, z) ((e s, z), (x, e s y)) r r 0 1 x Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 32 / 36

85 Constructing Embeddings Proof Sketch: Ψ : R J 1 M T R + T M (s, x, y, z) ((e s, z), (x, e s y)) r r 0 1 x Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 32 / 36

86 Constructing Embeddings Proof Sketch: Ψ : R J 1 M T R + T M (s, x, y, z) ((e s, z), (x, e s y)) r r 0 1 x Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 32 / 36

87 Width of Non-Cylindrical Lagrangian Cobordisms What if L is a non-cylindrical cobordism? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 33 / 36

88 Width of Non-Cylindrical Lagrangian Cobordisms What if L is a non-cylindrical cobordism? Example: Question: Can we still find a symplectic embedding of B(2)? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 33 / 36

89 Width of Non-Cylindrical Lagrangian Cobordisms What if L is a non-cylindrical cobordism? Example: Question: Can we still find a symplectic embedding of B(2)? Answer: Yes! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 33 / 36

90 Width of Non-Cylindrical Lagrangian Cobordisms What if L is a non-cylindrical cobordism? Example: Question: Can we still find a symplectic embedding of B(2)? Answer: Yes! Question: Can we embed a bigger ball? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 33 / 36

91 Width of Non-Cylindrical Lagrangian Cobordisms What if L is a non-cylindrical cobordism? Example: Question: Can we still find a symplectic embedding of B(2)? Answer: Yes! Question: Can we embed a bigger ball? Answer: No! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 33 / 36

92 Width of Non-Cylindrical Lagrangian Cobordisms What if L is a non-cylindrical cobordism? Example: Question: Can we still find a symplectic embedding of B(2)? Answer: Yes! Question: Can we embed a bigger ball? Answer: No! Width does not see the negative end! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 33 / 36

93 Upper Bound for Width of Lagrangian Cobordisms Theorem (Sabloff-T) If L is a Lagrangian cobordism from Λ to Λ + and Λ is fillable, then ( ) w L 0 2c(Λ + ), where c(λ + ) is the minimum fundamental capacity (for any augmentation). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 34 / 36

94 Upper Bound for Width of Lagrangian Cobordisms Theorem (Sabloff-T) If L is a Lagrangian cobordism from Λ to Λ + and Λ is fillable, then ( ) w L 0 2c(Λ + ), where c(λ + ) is the minimum fundamental capacity (for any augmentation). Proof is similar in spirit to the proof when L = R Λ: Use Seidel Isomorphism to get the existence of a J-holomorphic disk through ψ(0) ψ(b(α)). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 34 / 36

95 Length-Width Connection Can reprove our earlier length result between Λ = U(2) and Λ + = U(1): h ln 2 h 1 + s 0 2 _ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 35 / 36

96 Length-Width Connection Can reprove our earlier length result between Λ = U(2) and Λ + = U(1): h ln 2 h 1 + s 0 2 _ Corollary Suppose L is a Lagrangian cobordism from Λ = U(2) to Λ + = U(1) that is cylindrical outside [ h, 0]. Then ( ) ( ) 2 c(u(2)) ln 2 = ln = ln h. 1 c(u(1)) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 35 / 36

97 Length-Width Connection Can reprove our earlier length result between Λ = U(2) and Λ + = U(1): h ln 2 h 1 + s 0 2 _ Corollary Suppose L is a Lagrangian cobordism from Λ = U(2) to Λ + = U(1) that is cylindrical outside [ h, 0]. Then ( ) ( ) 2 c(u(2)) ln 2 = ln = ln h. 1 c(u(1)) Proof: 2e h c(u(2))=e h w((r Λ ) 0 ) = w((r Λ ) h ) w(l0 ) 2c(U(1)). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 35 / 36

98 Questions Much to be understood about length and width! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 36 / 36

99 Questions Much to be understood about length and width! Calculate widths of other Lagrangian cobordisms when Λ admits an augmentation/filling! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 36 / 36

100 Questions Much to be understood about length and width! Calculate widths of other Lagrangian cobordisms when Λ admits an augmentation/filling! Can we calculate the width or length of a Lagrangian cobordism when Λ does not admit an augmentation/filling? For example, when Λ is stabilized or loose? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 36 / 36

101 Questions Much to be understood about length and width! Calculate widths of other Lagrangian cobordisms when Λ admits an augmentation/filling! Can we calculate the width or length of a Lagrangian cobordism when Λ does not admit an augmentation/filling? For example, when Λ is stabilized or loose? Thank you! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 36 / 36