Strong Sard Conjecture for analyticsub-Riemannian structures in dimension 3

André Belotto

Aix-Marseille Université

Sub-Riemannian Geometry and Interactions

September 07, 2020

Collaborators

Alessio Figalli Adam Parusiński Ludovic Rifford

Preliminaries

In this talk, we consider:

A smooth connected manifold M of dimension n ≥ 3;A totally nonholonomic regular distribution ∆ of rank k < n;

A Riemannian metric g over M.

Local meaning: For every point x ∈ M, there is an openneighbourhood V where ∆ is locally parametrized by k linearlyindependent vector fields:

{X 1x , . . . ,X kx }

satisfying the Hörmander bracket generating condition

Lie{X 1x , · · · ,X kx

}(y) = TyM ∀y ∈ V.

During the seminar: V = M and ∆ = Span{X 1, . . . ,X k}.

Preliminaries

In this talk, we consider:

A smooth connected manifold M of dimension n ≥ 3;A totally nonholonomic regular distribution ∆ of rank k < n;

A Riemannian metric g over M.

Local meaning: For every point x ∈ M, there is an openneighbourhood V where ∆ is locally parametrized by k linearlyindependent vector fields:

{X 1x , . . . ,X kx }

satisfying the Hörmander bracket generating condition

Lie{X 1x , · · · ,X kx

}(y) = TyM ∀y ∈ V.

During the seminar: V = M and ∆ = Span{X 1, . . . ,X k}.

Horizontal paths

An absolutely continuous curve γ : [0, 1]→ M is called horizontalwith respect to ∆ if it satisfies

γ̇(t) ∈ ∆(γ(t)) for a.e. t ∈ [0, 1].

Trajectories of a control system: Fix x ∈ M and denote byx(·; x , u) : [0, 1]→ M the solution of:

ẋ(t) =k∑

i=1

ui (t)Xi (x(t)) for a.e. t ∈ [0, 1] and x(0) = x

where u = (u1, · · · , uk) ∈ Ux ⊂ L2([0, 1],Rk).

γ is horizontal ⇐⇒ it is a solution of the Cauchy problem.

Endpoint Mapping and singular controls

The End-Point Mapping from x is defined as

Ex : Ux −→ Mu 7−→ x(1; x , u).

A control u ∈ Ux ⊂ L2([0, 1],Rk) is said to be singular if it is acritical point of E x . We consider the set of singular controls

Sx = {u ∈ Ux ; u is singular} ⊂ L2([0, 1],Rk)

Consider the set of critical values of Ex :

X x := Ex (Sx) ⊂ M.

The Sard Conjecture

Sard’s Conjecture

For every x ∈ M, the set X x has Lebesgue measure zero.

Difficulty: In infinite dimension, Sard’s Theorem is known tofail (Bates and Moreira)

If dim(M) ≥ 4: Partial results for certain Carnot groups (ofprescribed ranks and steps):

Le Donne, Montgomery, Ottazzi, Pansu, and Vittone (2016).Boarotto and Vittone (2019)

The Sard Conjecture in dimension 3 (dimM = 3 andrank ∆ = 2)

The Martinet surface is the set:

Σ :={x ∈ M |∆(x) + [∆,∆](x) 6= TxM

},

Every singular horizontal path must be contained in Σ, that is:

X x ⊂ Σ, ∀x ∈ M.

Sard’s Conjecture in dimension 3

∀x ∈ M, the set X x has 2-dimensional Hausdorff measure zero.

Strong Sard’s Conjecture in dimension 3

∀x ∈ M, the sets X x have Hausdorff dimension at most one.

Main results when dimM = 3.

Theorem (Zelenko and Zhitomirskii, 1995)

Assume that ∆ is generic (in respect to the Whitney C∞-topology)than all the sets X x have Hausdorff dimension at most one.

Furthermore, whenever ∆ is generic, they prove that the Martinetsurface Σ is smooth.

Theorem (Belotto da Silva and Rifford, 2018)

Suppose that the Martinet surface Σ is smooth. Then for everyx ∈ M the set X x has 2-dimensional Hausdorff measure zero.

Main results when dimM = 3 and M and ∆ are analytic.

Theorem (Belotto da Silva and Rifford, 2018)

Under additional hypothesis on the singular set of Σ (which werecall later), for every x ∈ M the set X x has 2-dimensionalHausdorff measure zero.

Theorem (Belotto da Silva, Figalli, Parusinski and Rifford)

For every x ∈ M, the set X x is a countable union of semianalyticcurves and it has Hausdorff dimension at most 1.

Combining this result with (Hakavuori and Le Donne 2016) we get:

Corollary

Every singular minimizing geodesic γ : [0, 1]→ M is of class C 1.

Characteristic line foliation

Singular horizontal paths are tangent to TΣ and to ∆.

Lemma

Singular horizontal paths are concatenation of leaves (of finitelength) of the characteristic line foliation F .

Generic characteristic line foliation.

(Zelenko, Zhimtomirskii, 1995) shows that for ∆ generic:

(i) The Martinet surface Σ is smooth;

(ii) The singular points of F are isolated;

(iii) The singular points of F can be of the following types:

During the seminar: We will assume (i) and (ii).

Characteristic line foliation (local models)

We may consider the following two local models:

Vector-fields: We assume that ∆ = Span{∂x1 ; ∂x2 + A(x)∂x3}.

[X 1,X 2] = h(x)∂x3

and we conclude that:

Martinet : Σ = {h(x) = 0}Foliation :Z = X 1(h)X 2 − X 2(h)X 1.

One-form: We assume that ∆ = Span{µ} ⊂ Ω1M .

µ ∧ dµ = h · volM , iΣ : Σ→ M

and we conclude that:

Martinet : Σ = {h(x) = 0}Foliation : η = i∗Σ(µ).

Controlled divergence (preliminary)

Let S be a surface. We have the equivalence:

Z ←→ η if η = iZvolS .

and the divergence of Z may be computed from η:

dη = divS(Z) volS .

Example: if S = R2 and volS = dx ∧ dy :

Z = A(x , y)∂x + B(x , y)∂yη = A(x , y)dy − B(x , y)dx

And we easily verify that:

dη = (Ax + By )dx ∧ dy

Controlled divergence (key property)

Since the generator µ of ∆ is regular, so:

dµ = α ∧ µ+ h 〈µ, ∗µ〉−1 (∗µ)

so if i : S → Σ ⊂ M is a map (gen. max. rank) into Σ:

dη = i∗(dµ) = i∗(α ∧ µ) = i∗(α) ∧ η = i∗(α) ∧ (iZvolS)

This allows us to conclude that Z has controlled divergence:

divS(Z) ∈ Z(OS).

Example: Recalling that Z = A∂x + B∂y :

divS(Z)dx ∧ dy = (fdx + gdy) ∧ (Ady − Bdx)= (f · A + g · B)dx ∧ dy .

Blowing-up

For the talk: We consider polar coordinate changes

σ : S1 × R≥0 → R2(θ, r) 7→ (r cos(θ), r sin(θ))

and we denote E = σ−1(0) = S1 × {0} the exceptional divisor.

Formally: we work with blowings-up

σ : (S,E )→ (R2, 0)

where E = σ−1(0) = P1.

Elementary singular points.

A singularity p ∈ R2 of Z = A∂x + B∂y is elementary if:

Jac(Z) =[∂xA ∂yA∂xB ∂yB

]evaluated at p admits one eigenvalue with non-zero real part.

Example: Degenerate focus point

After performing two blowings-up, all singular points are saddles.

Reduction of singularities of characteristic foliations.

Theorem (Bendixson-Seidenberg)

Let S be an analytic smooth surface and F be a line foliation onS . There exists a proper analytic morphism

π : (S̃, Ẽ )→ (S,Sing(Z))

which is (locally) given by a finite composition of blowings-up,such that the strict transform F̃ of the line foliation only hasisolated elementary singularities.

For the characteristic line foliation, moreover:

(i) F̃ only has saddle points;

(ii) If Σ is smooth, F̃ is non-dicritical, i.e. tangent to Ẽ .

Monodromic convergent singular trajectories.

Proposition

If Z is analytic and p is a focus point, then either all trajectorieshave finite length, or all trajectories have infinite length.

Characteristic convergent singular trajectories.

Lemma

If there is one characteristic singular trajectory whose limit is thethe singular point p, then there are only a finite number of singulartrajectories with finite length (all of them characteristic), whoselimit is the singular point p.

Proof of Proposition: no singular point after blowing-up

We consider the return map T : Λ→ Λ.

Denote by γp,T (p) the trajectory between p and T (p) and:

L : Λ → Rp 7→ length(γp,T (p))

We can show that L is an analytic function and w.l.g. monotone.

length(γp) =∑n∈N

length(γT n(p),T n+1(p)) =⇒ length(γq) ≤ length(γp)

Proof of Proposition: general case

In general, we consider two types of transitions and length maps:

T1 :Λ1 → Λ2, T2 :Λ2 → Λ3L1 :Λ1 → R L2 :Λ2 → R

T2 belongs to a Hardy field (Speissegger, 2018);Combining with reduction of singularities of metrics:

∃K > 0, s.t. length(γq,T2(q)) ≤ K · length(γp,T2(p))

General case: singular Martinet surface

There are 2N distinct singular horizontal paths from z to x .

Resolution of singularities: If one path has finite length, all pathswith the same “jumps” have finite length.

Symplectic geometry: We obtain a contradiction.

Thank you for the attention!