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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

SUB-RIEMANNIAN METRIC SPACES

Let X =(X1, . . . Xr) be smooth linearly independent vector fields in Rn. A Lipschitz curve γ : [0, 1] → Rn is horizontal if ˙ γ(t) = r

j=1 hj(t)Xj(γ(t))

for a.e. t ∈ [0, 1]. Definition (CC distance) The Carnot-Carathéodory distance between x, y ∈ Rn is dc(x, y) := inf

• ℓ(γ) :=

ˆ 1 |h(t)|dt : γ : [0, 1] → Rn horizontal γ(0) = x, γ(1) = y

• .

If the bracket generating condition rank Lie{X1, . . . , Xr}(x) = n ∀ x ∈ Rn holds, then dc is a distance on Rn.

Sub-Riemannian geodesics Proof of the theorem

SUB-RIEMANNIAN METRIC SPACES

Let X =(X1, . . . Xr) be smooth linearly independent vector fields in Rn. A Lipschitz curve γ : [0, 1] → Rn is horizontal if ˙ γ(t) = r

j=1 hj(t)Xj(γ(t))

for a.e. t ∈ [0, 1]. Definition (CC distance) The Carnot-Carathéodory distance between x, y ∈ Rn is dc(x, y) := inf

• ℓ(γ) :=

ˆ 1 |h(t)|dt : γ : [0, 1] → Rn horizontal γ(0) = x, γ(1) = y

• .

If the bracket generating condition rank Lie{X1, . . . , Xr}(x) = n ∀ x ∈ Rn holds, then dc is a distance on Rn.

Sub-Riemannian geodesics Proof of the theorem

SUB-RIEMANNIAN METRIC SPACES

Let X =(X1, . . . Xr) be smooth linearly independent vector fields in Rn. A Lipschitz curve γ : [0, 1] → Rn is horizontal if ˙ γ(t) = r

j=1 hj(t)Xj(γ(t))

for a.e. t ∈ [0, 1]. Definition (CC distance) The Carnot-Carathéodory distance between x, y ∈ Rn is dc(x, y) := inf

• ℓ(γ) :=

ˆ 1 |h(t)|dt : γ : [0, 1] → Rn horizontal γ(0) = x, γ(1) = y

• .

If the bracket generating condition rank Lie{X1, . . . , Xr}(x) = n ∀ x ∈ Rn holds, then dc is a distance on Rn.

Sub-Riemannian geodesics Proof of the theorem

REGULARITY OF GEODESICS

A horizontal curve γ : [0, 1] → Rn is a geodesic if ℓ(γ) = dc(γ(0), γ(1)). Geodesics (locally) exist. Are they regular1? 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)

1When parametrized by arclength

Sub-Riemannian geodesics Proof of the theorem

REGULARITY OF GEODESICS

A horizontal curve γ : [0, 1] → Rn is a geodesic if ℓ(γ) = dc(γ(0), γ(1)). Geodesics (locally) exist. Are they regular1? 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)

1When parametrized by arclength

Sub-Riemannian geodesics Proof of the theorem

EXISTENCE OF TANGENT LINES

Theorem (Monti-Pigati-V.) Let γ : [−T, T] → Rn 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 (Rn, λdc) 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 (Rn, λdc)) 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

EXISTENCE OF TANGENT LINES

Theorem (Monti-Pigati-V.) Let γ : [−T, T] → Rn 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 (Rn, λdc) 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 (Rn, λdc)) 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

EXISTENCE OF TANGENT LINES

Theorem (Monti-Pigati-V.) Let γ : [−T, T] → Rn 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 (Rn, λdc) 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 (Rn, λdc)) 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

CARNOT GROUPS

Definition A Carnot (or stratified) group G is a connected, simply connected, nilpotent Lie group whose Lie algebra admits the stratification g = V1 ⊕ V2 ⊕ · · · ⊕ Vs where Vi = [V1, Vi−1], i = 2, . . . , s (s = “step”) and [V1, Vs] = {0}. We identify G ≡ g ≡ Rn by exponential coordinates exp(x1X1 + · · · + xnXn) ← → (x1, . . . , xn). We endow G with the sub-Riemannian structure induced by a left- invariant, bracket-generating basis X1, . . . , Xr of V1 (r = “rank”). The induced distance dc 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 ∈ V1.

Sub-Riemannian geodesics Proof of the theorem

CARNOT GROUPS

Definition A Carnot (or stratified) group G is a connected, simply connected, nilpotent Lie group whose Lie algebra admits the stratification g = V1 ⊕ V2 ⊕ · · · ⊕ Vs where Vi = [V1, Vi−1], i = 2, . . . , s (s = “step”) and [V1, Vs] = {0}. We identify G ≡ g ≡ Rn by exponential coordinates exp(x1X1 + · · · + xnXn) ← → (x1, . . . , xn). We endow G with the sub-Riemannian structure induced by a left- invariant, bracket-generating basis X1, . . . , Xr of V1 (r = “rank”). The induced distance dc 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 ∈ V1.

Sub-Riemannian geodesics Proof of the theorem

CARNOT GROUPS

Definition A Carnot (or stratified) group G is a connected, simply connected, nilpotent Lie group whose Lie algebra admits the stratification g = V1 ⊕ V2 ⊕ · · · ⊕ Vs where Vi = [V1, Vi−1], i = 2, . . . , s (s = “step”) and [V1, Vs] = {0}. We identify G ≡ g ≡ Rn by exponential coordinates exp(x1X1 + · · · + xnXn) ← → (x1, . . . , xn). We endow G with the sub-Riemannian structure induced by a left- invariant, bracket-generating basis X1, . . . , Xr of V1 (r = “rank”). The induced distance dc 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 ∈ V1.

Sub-Riemannian geodesics Proof of the theorem

EXCESS

Given a horizontal curve γ : [−T, T] → G and η > 0, the excess is Exc(γ, η) := inf

π hyperplane in V1

η

−η

dist(γ′(t), π)2 dt 1/2 . Main theorem - equivalent statement Let γ : [−T, T] → G be a geodesic parametrized by arclength. Then, there exists a sequence ηi → 0+ such that Exc(γ, ηi) → 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

EXCESS

Given a horizontal curve γ : [−T, T] → G and η > 0, the excess is Exc(γ, η) := inf

π hyperplane in V1

η

−η

dist(γ′(t), π)2 dt 1/2 . Main theorem - equivalent statement Let γ : [−T, T] → G be a geodesic parametrized by arclength. Then, there exists a sequence ηi → 0+ such that Exc(γ, ηi) → 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

EXCESS

Given a horizontal curve γ : [−T, T] → G and η > 0, the excess is Exc(γ, η) := inf

π hyperplane in V1

η

−η

dist(γ′(t), π)2 dt 1/2 . Main theorem - equivalent statement Let γ : [−T, T] → G be a geodesic parametrized by arclength. Then, there exists a sequence ηi → 0+ such that Exc(γ, ηi) → 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

STEP 1: CUT AND ERROR

Fact: upon identifying γ with log γ (a curve in g), a horizontal curve γ is uniquely determined by (γ(−T) and) its components γ in V1 ≡ Rr. Replace (“cut”)) γ |[−η,η] with a straight line. This produces a new curve γ(1) with length gain ℓ(γ) − ℓ(γ(1)) ≥ δ2 η. One now has to correct the error E = γ(T)−1 · γ(1)(T) = (0, E2, E3, . . . , Es) ∈ V2 ⊕ · · · ⊕ Vs with correction devices of length o(η). One first correct E2 producing γ(2) such that γ(T)−1 · γ(2)(T) = (0, 0, E′

3, . . . , E′ s) ∈ V3 ⊕ · · · ⊕ Vs.

Then one corrects E′

3, etc. This defines γ(3), . . . , γ(s).

Sub-Riemannian geodesics Proof of the theorem

STEP 1: CUT AND ERROR

Fact: upon identifying γ with log γ (a curve in g), a horizontal curve γ is uniquely determined by (γ(−T) and) its components γ in V1 ≡ Rr. Replace (“cut”)) γ |[−η,η] with a straight line. This produces a new curve γ(1) with length gain ℓ(γ) − ℓ(γ(1)) ≥ δ2 η. One now has to correct the error E = γ(T)−1 · γ(1)(T) = (0, E2, E3, . . . , Es) ∈ V2 ⊕ · · · ⊕ Vs with correction devices of length o(η). One first correct E2 producing γ(2) such that γ(T)−1 · γ(2)(T) = (0, 0, E′

3, . . . , E′ s) ∈ V3 ⊕ · · · ⊕ Vs.

Then one corrects E′

3, etc. This defines γ(3), . . . , γ(s).

Sub-Riemannian geodesics Proof of the theorem

STEP 1: CUT AND ERROR

Fact: upon identifying γ with log γ (a curve in g), a horizontal curve γ is uniquely determined by (γ(−T) and) its components γ in V1 ≡ Rr. Replace (“cut”)) γ |[−η,η] with a straight line. This produces a new curve γ(1) with length gain ℓ(γ) − ℓ(γ(1)) ≥ δ2 η. One now has to correct the error E = γ(T)−1 · γ(1)(T) = (0, E2, E3, . . . , Es) ∈ V2 ⊕ · · · ⊕ Vs with correction devices of length o(η). One first correct E2 producing γ(2) such that γ(T)−1 · γ(2)(T) = (0, 0, E′

3, . . . , E′ s) ∈ V3 ⊕ · · · ⊕ Vs.

Then one corrects E′

3, etc. This defines γ(3), . . . , γ(s).

Sub-Riemannian geodesics Proof of the theorem

STEP 1: CUT AND ERROR

Fact: upon identifying γ with log γ (a curve in g), a horizontal curve γ is uniquely determined by (γ(−T) and) its components γ in V1 ≡ Rr. Replace (“cut”)) γ |[−η,η] with a straight line. This produces a new curve γ(1) with length gain ℓ(γ) − ℓ(γ(1)) ≥ δ2 η. One now has to correct the error E = γ(T)−1 · γ(1)(T) = (0, E2, E3, . . . , Es) ∈ V2 ⊕ · · · ⊕ Vs with correction devices of length o(η). One first correct E2 producing γ(2) such that γ(T)−1 · γ(2)(T) = (0, 0, E′

3, . . . , E′ s) ∈ V3 ⊕ · · · ⊕ Vs.

Then one corrects E′

3, etc. This defines γ(3), . . . , γ(s).

Sub-Riemannian geodesics Proof of the theorem

CORRECTION DEVICES

Fact: after γ(k−1), one has to correct an error with leading part Ek ∈ Vk

• f size ∼ ηk(1−ǫk). Main obstruction:

dc(0, exp(Ek)) ∼ |Ek|1/k ∼ η1−ǫk ≫ η. Correction devices (Leonardi-Monti, Hakavuori-Le Donne) Given

◮ a horizontal curve c : [−T, T] → G ◮ [a, b] ⊂ [−T, T] ◮ Y ∈ Vj and a geodesic δY from 0 to exp(Y)

we define a corrected curve as the concatenation of (left-translations

• f) c|[−T,a], δY, c|[a,b], −δY and c|[b,T].

Key point 1: the final point is moved by an amount exp([Y, Z] + h.o.t.), where Z := c(b) − c(a).

Sub-Riemannian geodesics Proof of the theorem

CORRECTION DEVICES

Fact: after γ(k−1), one has to correct an error with leading part Ek ∈ Vk

• f size ∼ ηk(1−ǫk). Main obstruction:

dc(0, exp(Ek)) ∼ |Ek|1/k ∼ η1−ǫk ≫ η. Correction devices (Leonardi-Monti, Hakavuori-Le Donne) Given

◮ a horizontal curve c : [−T, T] → G ◮ [a, b] ⊂ [−T, T] ◮ Y ∈ Vj and a geodesic δY from 0 to exp(Y)

we define a corrected curve as the concatenation of (left-translations

• f) c|[−T,a], δY, c|[a,b], −δY and c|[b,T].

Key point 1: the final point is moved by an amount exp([Y, Z] + h.o.t.), where Z := c(b) − c(a).

Sub-Riemannian geodesics Proof of the theorem

CORRECTION DEVICES

Fact: after γ(k−1), one has to correct an error with leading part Ek ∈ Vk

• f size ∼ ηk(1−ǫk). Main obstruction:

dc(0, exp(Ek)) ∼ |Ek|1/k ∼ η1−ǫk ≫ η. Correction devices (Leonardi-Monti, Hakavuori-Le Donne) Given

◮ a horizontal curve c : [−T, T] → G ◮ [a, b] ⊂ [−T, T] ◮ Y ∈ Vj and a geodesic δY from 0 to exp(Y)

we define a corrected curve as the concatenation of (left-translations

• f) c|[−T,a], δY, c|[a,b], −δY and c|[b,T].

Key point 1: the final point is moved by an amount exp([Y, Z] + h.o.t.), where Z := c(b) − c(a).

Sub-Riemannian geodesics Proof of the theorem

STEP 2: APPLICATION OF CORRECTION DEVICES

Lemma Key point 2: if Exc(c, I) ≥ δ, then there exist a1 < b1 ≤ a2 < b2 ≤ · · · ≤ ar < br in I such that |det(Z1, Z2 . . . , Zr)| |I|r, Zi := c(bi) − c(ai) ∈ V1. In particular, |Zi| ∼ |I|. Apply the Lemma with I = [−ηαk, ηαk] and use Vk = [Vk−1, V1] to get Ek = [Y1, Z1] + · · · + [Yr, Zr], Yi ∈ Vk−1, |Yi| ∼ ηk(1−εk)−αk. Use now r correction devices placed at ai, bi pointing in direction −Yi. They move the final point by exp(−[Y1, Z1] − . . . − [Yr, Zr] + h.o.t.) = exp(−Ek + h.o.t.) and spend a length of order |Yi|

1 k−1 ∼ η1+βk ≪ η.

Sub-Riemannian geodesics Proof of the theorem

STEP 2: APPLICATION OF CORRECTION DEVICES

Lemma Key point 2: if Exc(c, I) ≥ δ, then there exist a1 < b1 ≤ a2 < b2 ≤ · · · ≤ ar < br in I such that |det(Z1, Z2 . . . , Zr)| |I|r, Zi := c(bi) − c(ai) ∈ V1. In particular, |Zi| ∼ |I|. Apply the Lemma with I = [−ηαk, ηαk] and use Vk = [Vk−1, V1] to get Ek = [Y1, Z1] + · · · + [Yr, Zr], Yi ∈ Vk−1, |Yi| ∼ ηk(1−εk)−αk. Use now r correction devices placed at ai, bi pointing in direction −Yi. They move the final point by exp(−[Y1, Z1] − . . . − [Yr, Zr] + h.o.t.) = exp(−Ek + h.o.t.) and spend a length of order |Yi|

1 k−1 ∼ η1+βk ≪ η.

Sub-Riemannian geodesics Proof of the theorem

STEP 2: APPLICATION OF CORRECTION DEVICES

Lemma Key point 2: if Exc(c, I) ≥ δ, then there exist a1 < b1 ≤ a2 < b2 ≤ · · · ≤ ar < br in I such that |det(Z1, Z2 . . . , Zr)| |I|r, Zi := c(bi) − c(ai) ∈ V1. In particular, |Zi| ∼ |I|. Apply the Lemma with I = [−ηαk, ηαk] and use Vk = [Vk−1, V1] to get Ek = [Y1, Z1] + · · · + [Yr, Zr], Yi ∈ Vk−1, |Yi| ∼ ηk(1−εk)−αk. Use now r correction devices placed at ai, bi pointing in direction −Yi. They move the final point by exp(−[Y1, Z1] − . . . − [Yr, Zr] + h.o.t.) = exp(−Ek + h.o.t.) and spend a length of order |Yi|

1 k−1 ∼ η1+βk ≪ η.

Sub-Riemannian geodesics Proof of the theorem

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