Sub-Riemannian geodesics Proof of the theorem Existence of tangent lines to sub-Riemannian geodesics joint with R. Monti and A. Pigati Davide Vittone Università di Padova Workshop on Geometric Measure Theory Warwick, July 11th, 2017
Sub-Riemannian geodesics Proof of the theorem S UB -R IEMANNIAN METRIC SPACES Let X =( X 1 , . . . X r ) be smooth linearly independent vector fields in R n . A Lipschitz curve γ : [ 0 , 1 ] → R n is horizontal if γ ( t ) = � r ˙ j = 1 h j ( t ) X j ( γ ( t )) for a.e. t ∈ [ 0 , 1 ] . Definition (CC distance) The Carnot-Carathéodory distance between x , y ∈ R n is ˆ 1 | h ( t ) | dt : γ : [ 0 , 1 ] → R n horizontal � � d c ( x , y ) := inf ℓ ( γ ) := . γ ( 0 ) = x , γ ( 1 ) = y 0 If the bracket generating condition ∀ x ∈ R n rank Lie { X 1 , . . . , X r } ( x ) = n holds, then d c is a distance on R n .
Sub-Riemannian geodesics Proof of the theorem S UB -R IEMANNIAN METRIC SPACES Let X =( X 1 , . . . X r ) be smooth linearly independent vector fields in R n . A Lipschitz curve γ : [ 0 , 1 ] → R n is horizontal if γ ( t ) = � r ˙ j = 1 h j ( t ) X j ( γ ( t )) for a.e. t ∈ [ 0 , 1 ] . Definition (CC distance) The Carnot-Carathéodory distance between x , y ∈ R n is ˆ 1 | h ( t ) | dt : γ : [ 0 , 1 ] → R n horizontal � � d c ( x , y ) := inf ℓ ( γ ) := . γ ( 0 ) = x , γ ( 1 ) = y 0 If the bracket generating condition ∀ x ∈ R n rank Lie { X 1 , . . . , X r } ( x ) = n holds, then d c is a distance on R n .
Sub-Riemannian geodesics Proof of the theorem S UB -R IEMANNIAN METRIC SPACES Let X =( X 1 , . . . X r ) be smooth linearly independent vector fields in R n . A Lipschitz curve γ : [ 0 , 1 ] → R n is horizontal if γ ( t ) = � r ˙ j = 1 h j ( t ) X j ( γ ( t )) for a.e. t ∈ [ 0 , 1 ] . Definition (CC distance) The Carnot-Carathéodory distance between x , y ∈ R n is ˆ 1 | h ( t ) | dt : γ : [ 0 , 1 ] → R n horizontal � � d c ( x , y ) := inf ℓ ( γ ) := . γ ( 0 ) = x , γ ( 1 ) = y 0 If the bracket generating condition ∀ x ∈ R n rank Lie { X 1 , . . . , X r } ( x ) = n holds, then d c is a distance on R n .
Sub-Riemannian geodesics Proof of the theorem R EGULARITY OF GEODESICS A horizontal curve γ : [ 0 , 1 ] → R n is a geodesic if ℓ ( γ ) = d c ( γ ( 0 ) , γ ( 1 )) . Geodesics (locally) exist. Are they regular 1 ? Main difficulty: geodesics can be abnormal curves (Montgomery ’94). Best results so far: ◮ geodesics in sub-Riemannian spaces of step 2 are smooth (Strichartz ’86-’89) ◮ geodesics are smooth in an open dense set (Sussmann ’15) ◮ geodesics have no corner-type singularities (Leonardi-Monti ’08, Le Donne-Leonardi-Monti-V. ’15, Hakavuori-Le Donne ’16) 1 When parametrized by arclength
Sub-Riemannian geodesics Proof of the theorem R EGULARITY OF GEODESICS A horizontal curve γ : [ 0 , 1 ] → R n is a geodesic if ℓ ( γ ) = d c ( γ ( 0 ) , γ ( 1 )) . Geodesics (locally) exist. Are they regular 1 ? Main difficulty: geodesics can be abnormal curves (Montgomery ’94). Best results so far: ◮ geodesics in sub-Riemannian spaces of step 2 are smooth (Strichartz ’86-’89) ◮ geodesics are smooth in an open dense set (Sussmann ’15) ◮ geodesics have no corner-type singularities (Leonardi-Monti ’08, Le Donne-Leonardi-Monti-V. ’15, Hakavuori-Le Donne ’16) 1 When parametrized by arclength
Sub-Riemannian geodesics Proof of the theorem E XISTENCE OF TANGENT LINES Theorem (Monti-Pigati-V.) Let γ : [ − T , T ] → R n be a geodesic parametrized by arclength. Then, the tangent cone Tan( γ, 0) contains a horizontal straight line. Fact 1: as λ → + ∞ , the metric space ( R n , λ d c ) converges to a (quo- tient of) a Carnot group G in the (pointed) Gromov-Hausdorff conver- gence (Mitchell, Margulis-Mostow). Tan( γ, 0) denotes the set of all possible limits of γ (which is a geodesic in ( R n , λ d c ) ) as λ → + ∞ , with base point γ ( 0 ) . Fact 2: all curves in Tan( γ, 0) are geodesics in G through the identity. Consequence: it is enough to prove the theorem for Carnot groups.
Sub-Riemannian geodesics Proof of the theorem E XISTENCE OF TANGENT LINES Theorem (Monti-Pigati-V.) Let γ : [ − T , T ] → R n be a geodesic parametrized by arclength. Then, the tangent cone Tan( γ, 0) contains a horizontal straight line. Fact 1: as λ → + ∞ , the metric space ( R n , λ d c ) converges to a (quo- tient of) a Carnot group G in the (pointed) Gromov-Hausdorff conver- gence (Mitchell, Margulis-Mostow). Tan( γ, 0) denotes the set of all possible limits of γ (which is a geodesic in ( R n , λ d c ) ) as λ → + ∞ , with base point γ ( 0 ) . Fact 2: all curves in Tan( γ, 0) are geodesics in G through the identity. Consequence: it is enough to prove the theorem for Carnot groups.
Sub-Riemannian geodesics Proof of the theorem E XISTENCE OF TANGENT LINES Theorem (Monti-Pigati-V.) Let γ : [ − T , T ] → R n be a geodesic parametrized by arclength. Then, the tangent cone Tan( γ, 0) contains a horizontal straight line. Fact 1: as λ → + ∞ , the metric space ( R n , λ d c ) converges to a (quo- tient of) a Carnot group G in the (pointed) Gromov-Hausdorff conver- gence (Mitchell, Margulis-Mostow). Tan( γ, 0) denotes the set of all possible limits of γ (which is a geodesic in ( R n , λ d c ) ) as λ → + ∞ , with base point γ ( 0 ) . Fact 2: all curves in Tan( γ, 0) are geodesics in G through the identity. Consequence: it is enough to prove the theorem for Carnot groups.
Sub-Riemannian geodesics Proof of the theorem C ARNOT GROUPS Definition A Carnot (or stratified ) group G is a connected, simply connected, nilpotent Lie group whose Lie algebra admits the stratification g = V 1 ⊕ V 2 ⊕ · · · ⊕ V s where V i = [ V 1 , V i − 1 ] , i = 2 , . . . , s ( s = “step”) and [ V 1 , V s ] = { 0 } . We identify G ≡ g ≡ R n by exponential coordinates exp ( x 1 X 1 + · · · + x n X n ) ← → ( x 1 , . . . , x n ) . We endow G with the sub-Riemannian structure induced by a left- invariant, bracket-generating basis X 1 , . . . , X r of V 1 ( r = “rank”). The induced distance d c is left-invariant. The tangent space to G is G itself. A horizontal line is a curve of the form t �→ exp ( tY ) for some Y ∈ V 1 .
Sub-Riemannian geodesics Proof of the theorem C ARNOT GROUPS Definition A Carnot (or stratified ) group G is a connected, simply connected, nilpotent Lie group whose Lie algebra admits the stratification g = V 1 ⊕ V 2 ⊕ · · · ⊕ V s where V i = [ V 1 , V i − 1 ] , i = 2 , . . . , s ( s = “step”) and [ V 1 , V s ] = { 0 } . We identify G ≡ g ≡ R n by exponential coordinates exp ( x 1 X 1 + · · · + x n X n ) ← → ( x 1 , . . . , x n ) . We endow G with the sub-Riemannian structure induced by a left- invariant, bracket-generating basis X 1 , . . . , X r of V 1 ( r = “rank”). The induced distance d c is left-invariant. The tangent space to G is G itself. A horizontal line is a curve of the form t �→ exp ( tY ) for some Y ∈ V 1 .
Sub-Riemannian geodesics Proof of the theorem C ARNOT GROUPS Definition A Carnot (or stratified ) group G is a connected, simply connected, nilpotent Lie group whose Lie algebra admits the stratification g = V 1 ⊕ V 2 ⊕ · · · ⊕ V s where V i = [ V 1 , V i − 1 ] , i = 2 , . . . , s ( s = “step”) and [ V 1 , V s ] = { 0 } . We identify G ≡ g ≡ R n by exponential coordinates exp ( x 1 X 1 + · · · + x n X n ) ← → ( x 1 , . . . , x n ) . We endow G with the sub-Riemannian structure induced by a left- invariant, bracket-generating basis X 1 , . . . , X r of V 1 ( r = “rank”). The induced distance d c is left-invariant. The tangent space to G is G itself. A horizontal line is a curve of the form t �→ exp ( tY ) for some Y ∈ V 1 .
Sub-Riemannian geodesics Proof of the theorem E XCESS Given a horizontal curve γ : [ − T , T ] → G and η > 0, the excess is � 1 / 2 � η dist ( γ ′ ( t ) , π ) 2 dt Exc ( γ, η ) := inf . π hyperplane in V 1 − η Main theorem - equivalent statement Let γ : [ − T , T ] → G be a geodesic parametrized by arclength. Then, there exists a sequence η i → 0 + such that Exc ( γ, η i ) → 0. The proof is by contradiction: assume there exists δ > 0 such that Exc ( γ, η ) > δ for any η > 0.
Sub-Riemannian geodesics Proof of the theorem E XCESS Given a horizontal curve γ : [ − T , T ] → G and η > 0, the excess is � 1 / 2 � η dist ( γ ′ ( t ) , π ) 2 dt Exc ( γ, η ) := inf . π hyperplane in V 1 − η Main theorem - equivalent statement Let γ : [ − T , T ] → G be a geodesic parametrized by arclength. Then, there exists a sequence η i → 0 + such that Exc ( γ, η i ) → 0. The proof is by contradiction: assume there exists δ > 0 such that Exc ( γ, η ) > δ for any η > 0.
Sub-Riemannian geodesics Proof of the theorem E XCESS Given a horizontal curve γ : [ − T , T ] → G and η > 0, the excess is � 1 / 2 � η dist ( γ ′ ( t ) , π ) 2 dt Exc ( γ, η ) := inf . π hyperplane in V 1 − η Main theorem - equivalent statement Let γ : [ − T , T ] → G be a geodesic parametrized by arclength. Then, there exists a sequence η i → 0 + such that Exc ( γ, η i ) → 0. The proof is by contradiction: assume there exists δ > 0 such that Exc ( γ, η ) > δ for any η > 0.
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