Equivalent Curves in Surfaces Anja Bankovi c University of - PowerPoint PPT Presentation

Equivalent Curves in Surfaces Anja Bankovi´ c University of Illinois

Equivalent Curves Fix a closed surface S , genus g ≥ 2.

Equivalent Curves Fix a closed surface S , genus g ≥ 2. (Horowitz, Randol) ∀ n ∈ Z + ∃ γ 1 , . . . , γ n , distinct homotopy classes of closed curves on S with ℓ m ( γ i ) = ℓ m ( γ j ) ∀ i , j ≤ n for every hyperbolic metric m on S .

Equivalent Curves Fix a closed surface S , genus g ≥ 2. (Horowitz, Randol) ∀ n ∈ Z + ∃ γ 1 , . . . , γ n , distinct homotopy classes of closed curves on S with ℓ m ( γ i ) = ℓ m ( γ j ) ∀ i , j ≤ n for every hyperbolic metric m on S . Problem: Do there exist pairs of distinct homotopy classes of curves γ and γ ′ which have the same length with respect to every metric in a given family of path metrics?

Equivalent Curves Fix a closed surface S , genus g ≥ 2. (Horowitz, Randol) ∀ n ∈ Z + ∃ γ 1 , . . . , γ n , distinct homotopy classes of closed curves on S with ℓ m ( γ i ) = ℓ m ( γ j ) ∀ i , j ≤ n for every hyperbolic metric m on S . Problem: Do there exist pairs of distinct homotopy classes of curves γ and γ ′ which have the same length with respect to every metric in a given family of path metrics? Generic metrics, probably ’no’...

Curves in Flat metrics Flat ( S , q ) = { metrics from q –differentials on S }

Curves in Flat metrics Flat ( S , q ) = { metrics from q –differentials on S } q -differential ϕ � p � q ζ ( p ) = ϕ ( z ) dz p 0 Preferred coordinates give an atlas of charts on S \{ zeros ( ϕ ) } to C

Curves in Flat metrics Flat ( S , q ) = { metrics from q –differentials on S } q -differential ϕ � p � q ζ ( p ) = ϕ ( z ) dz p 0 Preferred coordinates give an atlas of charts on S \{ zeros ( ϕ ) } to C C ( S ) = { closed curves on S } / homotopy

Curves in Flat metrics Flat ( S , q ) = { metrics from q –differentials on S } q -differential ϕ � p � q ζ ( p ) = ϕ ( z ) dz p 0 Preferred coordinates give an atlas of charts on S \{ zeros ( ϕ ) } to C C ( S ) = { closed curves on S } / homotopy Given γ, γ ′ ∈ C ( S ), define γ ≡ q γ ′ iff l m ( γ ) = l m ( γ ′ ), ∀ m ∈ Flat ( S , q )

Curves in Flat metrics Flat ( S , q ) = { metrics from q –differentials on S } q -differential ϕ � p � q ζ ( p ) = ϕ ( z ) dz p 0 Preferred coordinates give an atlas of charts on S \{ zeros ( ϕ ) } to C C ( S ) = { closed curves on S } / homotopy Given γ, γ ′ ∈ C ( S ), define γ ≡ q γ ′ iff l m ( γ ) = l m ( γ ′ ), ∀ m ∈ Flat ( S , q ) Theorem (C. Leininger) γ ≡ hyp γ ′ ⇒ γ ≡ 2 γ ′

Main results

Main results Theorem 1: ∀ q , k ∈ Z + , ∃ γ 1 , . . . , γ k ∈ C ( S ) such that γ i ≡ q γ j , ∀ i , j .

Main results Theorem 1: ∀ q , k ∈ Z + , ∃ γ 1 , . . . , γ k ∈ C ( S ) such that γ i ≡ q γ j , ∀ i , j . Define γ ≡ ∞ γ ′ iff l m ( γ ) = l m ( γ ′ ), ∀ m ∈ Flat ( S , q ) , ∀ q ∈ Z + .

Main results Theorem 1: ∀ q , k ∈ Z + , ∃ γ 1 , . . . , γ k ∈ C ( S ) such that γ i ≡ q γ j , ∀ i , j . Define γ ≡ ∞ γ ′ iff l m ( γ ) = l m ( γ ′ ), ∀ m ∈ Flat ( S , q ) , ∀ q ∈ Z + . Theorem 2: ≡ ∞ is trivial.

Main results Theorem 1: ∀ q , k ∈ Z + , ∃ γ 1 , . . . , γ k ∈ C ( S ) such that γ i ≡ q γ j , ∀ i , j . Define γ ≡ ∞ γ ′ iff l m ( γ ) = l m ( γ ′ ), ∀ m ∈ Flat ( S , q ) , ∀ q ∈ Z + . Theorem 2: ≡ ∞ is trivial. Theorem 3: γ ≡ 1 γ ′ ⇔ γ ≡ 2 γ ′

Euclidian Cone Metrics A metric σ on S is called a Euclidian cone metric if:

Euclidian Cone Metrics A metric σ on S is called a Euclidian cone metric if: (a) σ is a geodesic metric

Euclidian Cone Metrics A metric σ on S is called a Euclidian cone metric if: (a) σ is a geodesic metric (b) ∃ finite set X ⊂ S such that σ on S \ X is Euclidian.

Euclidian Cone Metrics A metric σ on S is called a Euclidian cone metric if: (a) σ is a geodesic metric (b) ∃ finite set X ⊂ S such that σ on S \ X is Euclidian. (c) ( ∀ x ∈ X )( ∃ ǫ > 0) B ǫ ( x ) is isometric to some cone.

Euclidian Cone Metrics A metric σ on S is called a Euclidian cone metric if: (a) σ is a geodesic metric (b) ∃ finite set X ⊂ S such that σ on S \ X is Euclidian. (c) ( ∀ x ∈ X )( ∃ ǫ > 0) B ǫ ( x ) is isometric to some cone.

Euclidian Cone Metrics A metric σ on S is called a Euclidian cone metric if: (a) σ is a geodesic metric (b) ∃ finite set X ⊂ S such that σ on S \ X is Euclidian. (c) ( ∀ x ∈ X )( ∃ ǫ > 0) B ǫ ( x ) is isometric to some cone. Let c ( x ) ∈ R + denote a cone angle around x .

Gluing surfaces A genus 2 surface with Euclidian cone metric:

Gluing surfaces A genus 2 surface with Euclidian cone metric:

Gluing surfaces A genus 2 surface with Euclidian cone metric:

Holonomy ∀ x ∈ S \ X , ρ x : π 1 ( S \ X , x ) → SO ( T x ( S \ X ))

Holonomy ∀ x ∈ S \ X , ρ x : π 1 ( S \ X , x ) → SO ( T x ( S \ X ))

Holonomy ∀ x ∈ S \ X , ρ x : π 1 ( S \ X , x ) → SO ( T x ( S \ X )) Gluing maps: { ρ i ◦ τ i } k i =1 . Hol ≤ � ρ 1 , ρ 2 , ..., ρ k � .

Flat metrics A cone metric is NPC iff c ( x ) ≥ 2 π , ∀ x ∈ S .

Flat metrics A cone metric is NPC iff c ( x ) ≥ 2 π , ∀ x ∈ S . Flat ( S ) = { m | m is NPC Euclidian cone metric on S inducing the given topology } .

Flat metrics A cone metric is NPC iff c ( x ) ≥ 2 π , ∀ x ∈ S . Flat ( S ) = { m | m is NPC Euclidian cone metric on S inducing the given topology } . ∀ q ∈ Z + , Flat ( S , q ) = { m ∈ Flat ( S ) | Hol ( m ) ⊂ � ρ 2 π q �} .

Geodesics in m ∈ Flat ( S ) ≥ π ≥ π ≥ π ≥ π ≥ π ≥ π

Geodesics in m ∈ Flat ( S ) ≥ π ≥ π ≥ π ≥ π ≥ π ≥ π Examples of geodesics on a genus 2 surface:

Proof of Theorem 1 Theorem 1: ∀ q , k ∈ Z + , ∃ γ 1 , . . . , γ k ∈ C ( S ) such that γ i ≡ q γ j , ∀ i , j .

Proof of Theorem 1 Theorem 1: ∀ q , k ∈ Z + , ∃ γ 1 , . . . , γ k ∈ C ( S ) such that γ i ≡ q γ j , ∀ i , j . Step 1: ∀ q ∈ Z + , ∃ γ ∈ C ( S ) so that ∀ m ∈ Flat ( S , q ) the geodesic γ m contains a cone point and [ γ ] � = 0 in H 1 ( S ).

Step 2: Build a rank-2 free group from this curve:

Step 2: Build a rank-2 free group from this curve: a b

Step 2: Build a rank-2 free group from this curve: a b

Step 2: Build a rank-2 free group from this curve: a b

Step 2: Build a rank-2 free group from this curve: a b

Step 2: Build a rank-2 free group from this curve: a b [ a ] = 2( q + 2)[ γ ] = [ b ]

Step 2: Build a rank-2 free group from this curve: a b [ a ] = 2( q + 2)[ γ ] = [ b ] w 0 = ( ab ) k , w 1 = ( ab ) k − 1 ( ab − 1 ) , . . . , w k − 1 = ( ab )( ab − 1 ) k − 1

Step 2: Build a rank-2 free group from this curve: a b [ a ] = 2( q + 2)[ γ ] = [ b ] w 0 = ( ab ) k , w 1 = ( ab ) k − 1 ( ab − 1 ) , . . . , w k − 1 = ( ab )( ab − 1 ) k − 1 [ w j ] = (2 k − 2 j )[ γ ]

Proof of Theorem 2 Theorem 2: ≡ ∞ is trivial.

Proof of Theorem 2 Theorem 2: ≡ ∞ is trivial. Step 1: ∀ γ, γ ′ ∈ C ( S ) , ∃ m ∈ Flat ( S ) so that l m ( γ ) � = l m ( γ ′ ).

Proof of Theorem 2 Theorem 2: ≡ ∞ is trivial. Step 1: ∀ γ, γ ′ ∈ C ( S ) , ∃ m ∈ Flat ( S ) so that l m ( γ ) � = l m ( γ ′ ).

Proof of Theorem 2 Theorem 2: ≡ ∞ is trivial. Step 1: ∀ γ, γ ′ ∈ C ( S ) , ∃ m ∈ Flat ( S ) so that l m ( γ ) � = l m ( γ ′ ). Step 2: Approximate the Flat ( S ) metrics by some metric in Flat ( S , q ) by appropriately choosing the metrics on the triangles.