Quantitative rectifiability in the Heisenberg group Katrin F assler - PowerPoint PPT Presentation

Quantitative rectifiability in the Heisenberg group Katrin F¨ assler University of Fribourg, Switzerland Joint work with Vasileios Chousionis and Tuomas Orponen

Quantitative rectifiability in the Heisenberg group (of codimension 1 sets) Katrin F¨ assler University of Fribourg, Switzerland Joint work with Vasileios Chousionis and Tuomas Orponen

The Heisenberg group H 1 = ( R 3 , ∗ ) d ( p , q ) = � q − 1 ∗ p � , where � ( x , y , t ) � = max {| ( x , y ) | , | t | 1 / 2 } The resulting space ( H 1 , d ) has Hausdorff dimension 4.

Rectifiability in H 1

Rectifiability in H 1 L. Ambrosio and B. Kirchheim, 2000 Let k ≥ 2. If f : A ⊆ R k → H 1 is Lipschitz, then H k ( f ( A )) = 0.

Rectifiability in H 1 L. Ambrosio and B. Kirchheim, 2000 Let k ≥ 2. If f : A ⊆ R k → H 1 is Lipschitz, then H k ( f ( A )) = 0. B. Franchi, R. Serapioni, and F. Serra Cassano, 2001–. . . In H 1 , consider an alternative to Federer’s definition of 3-rectifiability, for instance using intrinsic Lipschitz graphs .

Rectifiability in H 1 L. Ambrosio and B. Kirchheim, 2000 Let k ≥ 2. If f : A ⊆ R k → H 1 is Lipschitz, then H k ( f ( A )) = 0. B. Franchi, R. Serapioni, and F. Serra Cassano, 2001–. . . In H 1 , consider an alternative to Federer’s definition of 3-rectifiability, for instance using intrinsic Lipschitz graphs . Goal: study quantitative rectifiability in H 1 (for codimension 1) (low dimensional sets: Franchi-Ferrari-Pajot, Juillet, Li-Schul, Hahlomaa) Motivation: uniform rectifiability in R n (G. David and S. Semmes).

Towards the definition of intrinsic Lipschitz graphs Write H 1 = W ∗ V , where W - vertical 2-d subspace in R 3 , V - perpendicular 1-d horizontal subspace.

Towards the definition of intrinsic Lipschitz graphs Write H 1 = W ∗ V , where W - vertical 2-d subspace in R 3 , V - perpendicular 1-d horizontal subspace. Heisenberg projections π W : H 1 → W , p = p W ∗ p V �→ p W . Vertical projections: Horizontal projections: π V : H 1 → V , p = p W ∗ p V �→ p V .

Towards the definition of intrinsic Lipschitz graphs Write H 1 = W ∗ V , where W - vertical 2-d subspace in R 3 , V - perpendicular 1-d horizontal subspace. Heisenberg projections π W : H 1 → W , p = p W ∗ p V �→ p W . Vertical projections: Horizontal projections: π V : H 1 → V , p = p W ∗ p V �→ p V . Heisenberg cone C W ( α ) := { p ∈ H 1 : � π W ( p ) � ≤ α � π V ( p ) �} .

Intrinsic Lipschitz functions Intrinsic L -Lipschitz graph Γ ⊂ H 1 such that ( p ∗ C W ( 1 L )) ∩ Γ = { p } , for all p ∈ Γ .

Intrinsic Lipschitz functions Intrinsic L -Lipschitz graph Γ ⊂ H 1 such that ( p ∗ C W ( 1 L )) ∩ Γ = { p } , for all p ∈ Γ . Intrinsic L -Lipschitz function φ : A ⊆ W → V such that Γ φ := { w ∗ φ ( w ) : w ∈ A } is an intrinsic L -Lipschitz graph. ( φ is in general not metrically Lipschitz!)

The main result Theorem The following are equivalent for an Ahlfors 3 -regular E ⊂ H 1 : 1 big pieces of intrinsic Lipschitz graphs 2 big vertical projections + weak geometric lemma for vertical β ’s.

The main result Theorem The following are equivalent for an Ahlfors 3 -regular E ⊂ H 1 : 1 big pieces of intrinsic Lipschitz graphs 2 big vertical projections + weak geometric lemma for vertical β ’s. big pieces of intrinsic Lipschitz graphs : ∃ L ≥ 1 , θ > 0: ∀ x ∈ E , 0 < R ≤ diam ( E ) ∃ Γ intrinsic L -Lipschitz: H 3 ( E ∩ Γ ∩ B ( x , R )) ≥ θ R 3 .

The main result Theorem The following are equivalent for an Ahlfors 3 -regular E ⊂ H 1 : 1 big pieces of intrinsic Lipschitz graphs 2 big vertical projections + weak geometric lemma for vertical β ’s. big pieces of intrinsic Lipschitz graphs : ∃ L ≥ 1 , θ > 0: ∀ x ∈ E , 0 < R ≤ diam ( E ) ∃ Γ intrinsic L -Lipschitz: H 3 ( E ∩ Γ ∩ B ( x , R )) ≥ θ R 3 . big vertical projections : ∃ δ > 0: ∀ x ∈ E , 0 < R ≤ diam E ∃ W vertical: L 2 ( π W ( E ∩ B ( x , R ))) ≥ δ R 3

The main result Theorem The following are equivalent for an Ahlfors 3 -regular E ⊂ H 1 : 1 big pieces of intrinsic Lipschitz graphs 2 big vertical projections + weak geometric lemma for vertical β ’s. big pieces of intrinsic Lipschitz graphs : ∃ L ≥ 1 , θ > 0: ∀ x ∈ E , 0 < R ≤ diam ( E ) ∃ Γ intrinsic L -Lipschitz: H 3 ( E ∩ Γ ∩ B ( x , R )) ≥ θ R 3 . big vertical projections : ∃ δ > 0: ∀ x ∈ E , 0 < R ≤ diam E ∃ W vertical: L 2 ( π W ( E ∩ B ( x , R ))) ≥ δ R 3 dist ( y , z ∗ W ) vertical β -numbers : β ( B ) := inf W , z sup y ∈ B ∩ E . rad ( B )

The main result Theorem The following are equivalent for an Ahlfors 3 -regular E ⊂ H 1 : 1 big pieces of intrinsic Lipschitz graphs 2 big vertical projections + weak geometric lemma for vertical β ’s. big pieces of intrinsic Lipschitz graphs : ∃ L ≥ 1 , θ > 0: ∀ x ∈ E , 0 < R ≤ diam ( E ) ∃ Γ intrinsic L -Lipschitz: H 3 ( E ∩ Γ ∩ B ( x , R )) ≥ θ R 3 . big vertical projections : ∃ δ > 0: ∀ x ∈ E , 0 < R ≤ diam E ∃ W vertical: L 2 ( π W ( E ∩ B ( x , R ))) ≥ δ R 3 dist ( y , z ∗ W ) vertical β -numbers : β ( B ) := inf W , z sup y ∈ B ∩ E . rad ( B ) weak geometric lemma : ∀ ε > 0 , x ∈ E , R > 0 � R � χ { ( y , s ) ∈ E × R + : β ( B ( y , s )) ≥ ε } ( y , s ) d H 3 ( y ) d s s � ε R 3 . 0 E ∩ B ( x , R )

Properties of intrinsic Lipschitz functions B. Franchi, R. Serapioni, F. Serra Cassano (2011); A. Naor, R. Young (2017) : Properties of intrinsic L -Lipschitz φ : A ⊆ W → V

Properties of intrinsic Lipschitz functions B. Franchi, R. Serapioni, F. Serra Cassano (2011); A. Naor, R. Young (2017) : Properties of intrinsic L -Lipschitz φ : A ⊆ W → V p ∗ Γ φ is intrinsic L -Lipschitz ∀ p ∈ H 1

Properties of intrinsic Lipschitz functions B. Franchi, R. Serapioni, F. Serra Cassano (2011); A. Naor, R. Young (2017) : Properties of intrinsic L -Lipschitz φ : A ⊆ W → V p ∗ Γ φ is intrinsic L -Lipschitz ∀ p ∈ H 1 φ can be extended to an L ′ ( L )-intrinsic Lipschitz function W → V

Properties of intrinsic Lipschitz functions B. Franchi, R. Serapioni, F. Serra Cassano (2011); A. Naor, R. Young (2017) : Properties of intrinsic L -Lipschitz φ : A ⊆ W → V p ∗ Γ φ is intrinsic L -Lipschitz ∀ p ∈ H 1 φ can be extended to an L ′ ( L )-intrinsic Lipschitz function W → V Γ φ is Ahlfors-3-regular with constant depending on L (for A = W )

Properties of intrinsic Lipschitz functions B. Franchi, R. Serapioni, F. Serra Cassano (2011); A. Naor, R. Young (2017) : Properties of intrinsic L -Lipschitz φ : A ⊆ W → V p ∗ Γ φ is intrinsic L -Lipschitz ∀ p ∈ H 1 φ can be extended to an L ′ ( L )-intrinsic Lipschitz function W → V Γ φ is Ahlfors-3-regular with constant depending on L (for A = W ) φ is intrinsically differentiable in L 2 a.e. w ∈ A , with intrinsic differential d φ w : W → V , d φ w ( y , t ) = ∇ φ φ ( w ) y , Intrinsically differentiable : (for φ (0) = 0) ∃ d φ 0 : W → V , intrinsic linear: | φ ( y , t ) − d φ 0 ( y , t ) | = o ( � ( y , t ) � ); the general definition is by left-translation of the graph

Properties of intrinsic Lipschitz functions B. Franchi, R. Serapioni, F. Serra Cassano (2011); A. Naor, R. Young (2017) : Properties of intrinsic L -Lipschitz φ : A ⊆ W → V p ∗ Γ φ is intrinsic L -Lipschitz ∀ p ∈ H 1 φ can be extended to an L ′ ( L )-intrinsic Lipschitz function W → V Γ φ is Ahlfors-3-regular with constant depending on L (for A = W ) φ is intrinsically differentiable in L 2 a.e. w ∈ A , with intrinsic differential d φ w : W → V , d φ w ( y , t ) = ∇ φ φ ( w ) y , φ has bounded intrinsic gradient �∇ φ φ � ∞ ≤ L

Properties of intrinsic Lipschitz functions B. Franchi, R. Serapioni, F. Serra Cassano (2011); A. Naor, R. Young (2017) : Properties of intrinsic L -Lipschitz φ : A ⊆ W → V p ∗ Γ φ is intrinsic L -Lipschitz ∀ p ∈ H 1 φ can be extended to an L ′ ( L )-intrinsic Lipschitz function W → V Γ φ is Ahlfors-3-regular with constant depending on L (for A = W ) φ is intrinsically differentiable in L 2 a.e. w ∈ A , with intrinsic differential d φ w : W → V , d φ w ( y , t ) = ∇ φ φ ( w ) y , φ has bounded intrinsic gradient �∇ φ φ � ∞ ≤ L

Proof: Intrinsic Lipschitz graph ⇒ WGL Let Γ = Γ φ be the graph of an intrinsic Lipschitz function φ over W .

Proof: Intrinsic Lipschitz graph ⇒ WGL Let Γ = Γ φ be the graph of an intrinsic Lipschitz function φ over W . First attempt to prove WGL for Γ Γ ∩ B ( y , s ) is far from a vertical plane ? ⇒ gradient of φ fluctuates significantly in Γ ∩ B ( y , s ) ⇒ (with L ∞ bound for gradient) WGL.

Proof: Intrinsic Lipschitz graph ⇒ WGL Let Γ = Γ φ be the graph of an intrinsic Lipschitz function φ over W . First attempt to prove WGL for Γ Γ ∩ B ( y , s ) is far from a vertical plane ? ⇒ gradient of φ fluctuates significantly in Γ ∩ B ( y , s ) ⇒ (with L ∞ bound for gradient) WGL. This argument does not work for the intrinsic gradient!