Conformality and Q harmonicity in sub-Riemannian manifolds Joint - - PowerPoint PPT Presentation

Conformality and Q−harmonicity in sub-Riemannian manifolds

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Paris, September 29th October 3rd, 2014 Geometric Analysis on sub-Riemannian manifolds Institut Henri Poincar´ e

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

What is this talk about?

◮ Our goal is to prove smoothness of quasiconformal mappings

with minimal distortion (i.e. 1−quasiconformal mappings) between domains of subRiemannian manifolds out of regularity theory for sub elliptic p−laplacians.

◮ This implies that 1−quasiconformal mappings are conformal

diffeomorphisms.

◮ The proof is based on nonlinear subelliptic PDE, techniques

from analysis in metric spaces and from subRiemannan geometry.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

SubRiemannian manifolds

A subRiemannian manifold as a triplet (M, ∆, g) where

◮ M is a connected, smooth manifold of dimension n ∈ N, ◮ ∆ denotes a subbundle of the tangent bundle TM that

bracket generates TM,

◮ g is a positive definite smooth, bilinear form defined on ∆.

Iteratively set ∆1 := ∆, and ∆i+1 := ∆i + [∆i, ∆] for i ∈ N.The bracket generating condition (also called H¨

• rmander’s finite rank

hypothesis) is expressed by the existence of m ∈ N such that, for all p ∈ M, one has ∆m

p = TpM.

(1)

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

SubRiemannian manifolds

Analogously to the Riemannian setting, one can endow (M, ∆, g) with a metric space structure by defining the Carnot-Caratheodory (CC) control distance: For any pair x, y ∈ M set d(x, y) = inf{δ > 0 such that there exists a curve γ ∈ C ∞([0, 1]; M) with endpoints x, y such that ˙ γ ∈ ∆(γ) and |˙ γ|g ≤ δ}.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

SubRiemannian manifolds

Definition

A subRiemannian manifold (M, ∆, g) is equiregular if, for all i ∈ N, the dimension of ∆i

p is constant in p ∈ M. In this case, the

homogenous dimension is Q :=

m−1

• i=1

i [dim(∆i+1

p

) − dim(∆i

p)].

(2) Consider the metric space (M, d) where (M, ∆, g) is an equiregular subRiemannian manifold and d is the corresponding control metric. As a consequence of Chow-Rashevsky Theorem such a distance is always finite and induces on M the original topology. As a result of Mitchell, the Hausdorff dimension of (M, d) coincide with (2).

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Quasiconformal mappings

Let f : X → Y be a continuous map between two geodesics metric spaces and set for all x ∈ X Hf (x) := lim sup

r→0

supd(x,x′)≤r d(f (x), f (x′)) infd(x,x′)≥rd(f (x), f (x′)) ,

◮ If K ≥ 1 then a homeomorphism f : X → Y is K−QCF if

Hf (x) ≤ K at any point x ∈ X.

◮ This class arose in connection with a L∞ extremal problem

(1928) and provides a relaxation of the notion of conformality. In particular, for K = 1 one expects a synthetic notion of conformality, i.e. infinitesimal circles are mapped into infinitesimal circles.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Quasiconformal mappings

◮ Given a compatible measure structure (i.e. doubling +

Poincar´ e), there is an equivalent analytic definition f ∈ W 1,n

loc (X) and

(Lipf )Q ≤ KJf Here Jf (x) := lim

r→0 |f (B(x, r))|/|B(x, r)|

and Lipf (x) = lim sup

p→x

d(f (p), f (x))/d(x, p).

◮ This class is closed under uniform convergence

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

1−Quasiconformal mappings: Some history

• Euclidean , n ≥ 3 (1850, Liouville) C 3 conformal diffeomorphisms

are finite dimensional.

• Euclidean , n ≥ 2 (1958, Hartman) C 1 conformal diffeo. are

smooth.

• Euclidean , n ≥ 2 (1963, Gehring and Reshetnyak) 1−QCF maps

are conformal diffeo (De Giorgi-Nash-Moser theorem)

• Riemannian (1976, Ferrand) 1−QCF maps are conformal diffeo.

(after Reshetnyak). (Myers-Steenrod, Sharp Isoperimetric Ineq.)

• Riemannian (2013, Liimatainen and Salo) new proof, based on

n−harmonic coordinates (after Taylor’s approach to Myers-Steenrod).

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Sketch of the proof in the Riemannian setting

Goal: Show smoothness of 1−QCF maps f : (M, g) → (N, h). A PDE for conformal energy A function u ∈ W 1,n(M, dµ) ( dµ Riemannian volume) is n−harmonic if

• N |∇hu|n−2∇hu∇hφdµ = 0

for φ ∈ C ∞

0 (N).

• Step 1 1−QCF are differentiable a.e. and |df |n = Jf .
• Step 2 1−QCF are Lipschitz.
• Step 3 If u is n−harmonic then so is u ◦ f . (Morphism Property)
• Step 4 There exists n−harmonic coordinates x1, ..., xn near every

point in N

• Step 5 The functions fi = xi ◦ f are n−harmonic in M and |∇gfi|

is bnd away from zero and ∞.

• Step 6 Regularity for p−Laplacian yields smoothness.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

subRiemannian Setting

The introduction of conformal and quasiconformal maps in the subRiemannian setting goes back to the proof of Mostow’s rigidity theorem, where such maps arise as boundary limits of quasi-isometries between certain Gromov hyperbolic spaces. In view of work by Koranyi, Reimann, Pansu, Tang, Cowling, Heinonen, Koskela, etc. etc... Theorem If X, Y are Carnot groups, and f is an homeomorphism with Hf (x) = 1 identically then f is smooth and its differential Tf is a similarity with non-zero dilation ratio at every point.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Does the result continue to hold in the setting of SubRiemannian manifolds?

Denote by Q the homogeneous dimension of (M, ∆, g). Set d vol a smooth volume form and for u ∈ W 1,Q

H,loc(M), define the

Q-Laplacian LQu by means of the following identity

• M

LQuφd vol :=

• M

|∇Hu|Q−2∇Hu, ∇Hφgd vol, for any φ ∈ W 1,Q (M)). If u ∈ W 2,2

H,loc(M) ∩ W 1,Q H,loc(M) one can

then write LQu = X ∗

i (|∇Hu|Q−2Xiu) a.e. in M.

Definition (Q-harmonic function)

Let M be an equiregular sub-Riemannian manifold of Hausdorff dimension Q. Fixed a measure vol on M, a function u ∈ W 1,Q

H,loc(M) is called Q-harmonic if

• M

|∇Hu|Q−2∇Hu, ∇Hφ d vol = 0, ∀φ ∈ W 1,Q

H,0 (M, vol).

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Does the result continue to hold in the setting of subRiemannian manifolds?

Definition

Consider a subRiemannian manifold M endowed with a smooth volume form vol. We say that M supports regularity for Q-harmonic functions if for each Q-harmonic function u the following two properties hold:

• 1. For every U ⊂⊂ M there exist constants α ∈ (0, 1), C > 0

depending on ||u||W 1,Q, U, Q, g, d vol, such that ||u||C 1,α

H

(U) ≤ C.

(3)

• 2. For any domain K ⊂ M where |∇Hu| is bounded strictly away

from zero, there exists a constant C > 0 depending on ||u||W 1,Q, K, Q, g, d vol, such that ||u||W 2,2

H (K,dvol) ≤ C

(4)

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Does the result continue to hold in the setting of SubRiemannian manifolds?

Theorem

Our result: If M, N support C 1,α regularity for Q−harmonic scalar functions then 1−QCF maps f : M → N are smooth. Ongoing work SubRiemannian contact manifolds support C 1,α regularity for Q−harmonic scalar functions. Remark The only case known where C 1,α regularity for Q−harmonic scalar functions is currently known is the Heisenberg group endowed with a left-invariant metric. (Zhong 2008). this is a hard problem with a long list of partial results by many authors (Domokos, Manfredi, Marchi, Mingione, Zatorska-Goldstein,...).

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Tangent Cones

The tangent cone of a metric space (M, d) at a point x ∈ M is TxM := limλ→∞(M, λd), with x chosen as base point for all spaces (M, λd).

Theorem (Mitchell 1985)

For an equiregular subRiemannian manifold (M, ∆, g), the tangent cone at x ∈ M is isometric to Nx(M), with Nx(M) a Carnot group The result had been previously established by Pansu for nilpotent Lie groups.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Mostow-Margulis Differential

Theorem (Mostow-Margulis, 1995)

Let f : M → N be a quasi-conformal map between equiregular sub-Riemannian manifolds M and N, and let fλ : (M, λd) → (N, λd) denote a continuous map induced by f . The for almost every p ∈ M, f is CC differentiable at p; the maps fλ converge uniformly on compacts sets as λ → ∞ to a continuous map Np(f ) : Np(M) → Nf (p)(N) a group isomorphism that commutes with the group dilations. This extends earlier results of Pansu, who studied the nilpotent Lie group case.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Jacobians

Let (M1, µ1) and (M2, µ2) be metric measure spaces and f : M1 → M2 a homeomorphism. We say that Jf : M1 → R is a Jacobian for f with respect to the measures µ1 and µ2, if f ∗µ2 = Jf µ1, which is equivalent to the change of variable formula:

• f (A)

hδµ2 =

• A

(h ◦ f ) Jf δµ1, for every A ⊂ M1 measurable and every continuous function h : M2 → R.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

A result of Agrachev-Barilari-Boscain and Ghezzi-Jean

Given M an equiregular subRiemannian manifold of Hausdorff dimension Q, we need to work with the Popp measure volM rather than the spherical Hausdorff measure SQ

M since volM is always

smooth whereas there are cases in which SQ

M is not .

One has the following formula d volM = 2−Q volNp(M)(BNp(M)(e, 1))dSQ

M,

(5) where we used the fact that the measure induced on Np(M) by volM is volNp(M). We also rely on the following result from Ghezzi-Jean, volM(B(q, ǫ)) = ǫQvolNq(M)(BNq(M)(e, 1)) + o(ǫQ). (6)

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Our results: Equivalent notions of conformality

By 1−quasiconformal mapping in the subRiemannian setting we intend an homeomorphism f : M → N with Hf (x) = 1 identically. Lemma If f : M → N is quasiconformal then the following are equivalent (i) Hf = 1 (ii) At every point of differentiability HNp(f ) = 1 where Np(f ) : NpM → NpN is Mostow-Margulis differential of f at p ∈ M. (iii) Lipf = lipf = LipNp(f ) = lipNp(f ) at every point of

• differentiability. Here we have set

Lipf (x) = lim sup

y→x

d(f (x), f (y)) d(x, y) and lf (x) = lim inf

y→x

d(f (x), f (y)) d(x, y) (iv) At every point of differentiability LipQ

f = Jpopp f

= JHaus

f

.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Sketch of: Lipf = lf implies LipQ

f = Jpopp f

.

If f : M → N is 1−QCF then one can show ultra-filters in the Margulis-Mostow’s constructions to show that Np(f ) : NpM → NpN is a similarity. Consequently NpM and NpN are isometric Consequently they have the same Popp measures volNpM(B(NpM(1))) = volNf (p)N(B(Nf (p)N(1)))

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Sketch of: Lipf = lf implies LipQ

f = Jpopp f

.

Consequently, applying the results of Ghezzi-Jean, Agrachev-Barilari-Boscain volM(B(q, ǫ)) = ǫQvolNq(M)(BNq(M)(e, 1)) + o(ǫQ),

• ne obtains (for every ε > 0)

volN(f (B(p, r))) volM(B(p, r)) ≤ volN(B(f (p), r(Lipf (p) + ǫ))) volM(B(p, r)) =

• volNf (p)N(BNf (p)N(1))
• (Lipf (p) + ǫ)Q

volNp(M)(BNp(M)(1)) + o(1) Hence, as ε, r → 0 one obtains Jf (p) ≤ LipQ

f (p). The opposite

inequality is proved by considering f −1.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Our results: Morphism property

f is 1−qcf and u ∈ W 1,Q

loc (N) then

u := u ◦ f ∈ W 1,Q

loc (M) and

• K ′ |∇0u|Q(y)dVolP(y) =
• K

|∇ u|Q(x)dVolP(x), for any K ′ = f (K) ⊂⊂ N.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

In particular, if g, u ∈ C ∞(Y ) satisfy

• Y

|∇0u|Q−2∇0u∇0φdVol(y) =

• Y

g(y)φ(y)dVol(y) then u = u ◦ f and φ = φ ◦ f satisfy

• X

|∇0 u|Q−2∇0 u∇0 φdVol(x) =

• X

[g(f (x))λ−Q] φ(x)dVol(x) The presence of g creates problems in this nonlinear PDE

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Our results: Horizontal harmonic coordinates

Definition

Let M be a sub-Riemannian manifold. Let x1, . . . , xn be a system

• f coordinates on an open set U of M and let X1, . . . , Xr be a

frame of the horizontal distribution on U. We say that x1, . . . , xr are horizontal coordinates with respect to X1, . . . , Xr if the matrix (Xixj)(p), with i, j = 1, . . . , r, is invertible, for every p ∈ U. It is clear that any system of coordinates x1, . . . ., xn around a point p ∈ M can be reordered so that the first r components become a system of horizontal coordinates.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Our results: Horizontal harmonic coordinates

Theorem

Let M be an equiregular subRiemannian structure endowed with a smooth volume form vol. For any point p ∈ M there exists a set of horizontal harmonic coordinates defined in a neighborhood of p.

Theorem

Let M be an equiregular subRiemannian structure endowed with a smooth volume form vol that supports regularity for Q-harmonic

• functions. For any point p ∈ M there exists a set of horizontal

coordinates defined in a neighborhood of p that are Q-harmonic.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

How to construct horizontal harmonic coordinates

• Step 1 Considering any system of coordinates x1, . . . , xn in a

neighborhood of p ∈ M. Without loss of generality we can assume that the vectors ∇Hx1, . . . , ∇Hxr are linearly independent in a neighborhood of p, i.e., x1, .., xr are horizontal coordinates.

• Step 2 Solve the Dirichlet Problem
• L2ui

ǫ = 0 in Bǫ, i = 1, . . . , n

ui

ǫ = xi in ∂Bǫ, i = 1, . . . , n.

We show that for ε > 0 sufficiently small, the n-tuple u1

ε, . . . , ur ε, xr+1, . . . , xn is a system of coordinates. Note that

u1

ε, . . . , un ε may fail to be a system of coordinates.

• Step 3 H¨
• rmander’s hypoellipticity result yields

ui

ǫ ∈ C ∞(Bǫ) ∩ W 1,2 H (Bǫ). Consider now

wi

ǫ := ui ǫ − xi ∈ C ∞(Bǫ) ∩ W 1,2 H,0(Bǫ).

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

How to construct horizontal harmonic coordinates

• Step 4 Prove that

−

• Bǫ

|∇Hwi

ǫ|2 d vol ≤ C ′ǫ2,

and use Schauder estimates to get wC 1,α

H

(Bǫ/2) ≤ C.

• Step 5 Interpolate the L2 and the C α estimates to obtain L∞

estimates sup

B ǫ

4

|∇Hui

ǫ − ∇Hxi| ≤ o(1)

as ǫ → 0.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

... at this point

If f : M → N is a 1− quasiconformal mapping and the coordinates x1, ..., xn near f (p) ∈ N are a system of Q−harmonic horizontal coordinates, then x1 ◦ f , ..., xr ◦ f are Q−harmonic, and hence if the regularity theory holds they are in C 1,α ∩ W 2,2. What can we say about the rest of the coordinates?

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Our results: Horizontal regularity implies smoothness

Let x1, . . . , xn be coordinates on M such that x1, . . . , xr are horizontal coordinates with respect to an horizontal frame X1, . . . , Xr.

Proposition

Let M and N two sub-Riemannian manifolds. Let f : M → N be an ACC map. Let k ≥ 1, α ∈ (0, 1), and p ≥ 1. If f 1, . . . , f r are in C k,α

H,loc(M) (resp. in W k,p H,loc(M)), then f 1, . . . , f n is C k,α H,loc(M)

(resp. in W k,p

H,loc(M)).

Here we recall that ACC maps are those that send almost every (with respect to the Q-modulus measure) rectifiable curve into a rectifiable curve.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Some PDE remarks: Beyond C 1,α regularity

For every function u ∈ C 1,α

H

∩ W 2,2

H,loc(M), a.e. in M, the

Q-Laplacian can be expressed in non-divergence form LQu = aij(∇Hu)XiXju + g(x, ∇Hu), (8) where aij(ξ) = η−1

• |ξ|Q−4(δij + (Q − 2))ξiξj
• and

g(x, ξ) = η−1

• X ∗

i η|ξ|Q−2ξi + ∂kbi k(x)aij(ξ)ξk.

• .

The regularity hypothesis yields aij(∇Hu) and g(x, ∇Hu) ∈ C α

H,loc(M).

In view of the non-vanishing of ∇Hu one can invoke Schauder theory to boost up regularity to C ∞.

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

Coda: Sketch of Ferrand’s proof

σ(∂f (Br)) =

• f (∂Br)

fdσ ≤

• ∂Br

LipQ−1

f

dσ ≤

∂Br

LipQ

f dσ

(Q−1)/Q σ(∂Br)1/Q =

∂Br

JQ

f dσ

(Q−1)/Q σ(∂Br)1/Q ≤

• d

dr

• Br

Jf dVol (Q−1)/Q σ(∂Br)1/Q = d dr Vol(f (Br))(Q−1)/Qσ(∂Br)1/Q

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds

On the other hand ...

Isoperimetric Inequality Vol(f (Br)) ≤ C −1/(Q−1)

iso

σ(∂f (Br))Q/(Q−1) with Ciso = QQVol(B1). Yielding Vol(f (Br)) ≤ C −1/(Q−1)

iso

d dr Vol(f (Br)) σ(∂Br)1/(Q−1) = r Q d dr Vol(f (Br)) Hence Vol(f (Br)) rQ ≤ C i.e. Lipf is bounded. The fact that the isoperimetric profile of the Euclidean space is given by metric balls plays a crucial role

Joint work L.C., Enrico Le Donne (Jyv¨ askyl¨ a) and Alessandro Ottazzi (CIRM, Trento) Conformality and Q−harmonicity in sub-Riemannian manifolds