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¨ ormander’s finite rank hypothesis ) is expressed by the existence of m ∈ N such that, for all p ∈ M , one has ∆ m p = T p M . (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 m − 1 � i [dim(∆ i +1 ) − dim(∆ i Q := p )] . (2) p i =1 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 sup d ( x , x ′ ) ≤ r d ( f ( x ) , f ( x ′ )) H f ( x ) := lim sup inf d ( x , x ′ ) ≥ r d ( f ( x ) , f ( x ′ )) , r → 0 ◮ If K ≥ 1 then a homeomorphism f : X → Y is K − QCF if H f ( 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 ≤ � KJ f Here r → 0 | f ( B ( x , r )) | / | B ( x , r ) | J f ( x ) := lim and Lip f ( x ) = lim sup d ( f ( p ) , f ( x )) / d ( x , p ) . p → x ◮ 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 , d µ ) ( d µ � N |∇ h u | n − 2 ∇ h u ∇ h φ d µ = 0 Riemannian volume) is n − harmonic if for φ ∈ C ∞ 0 ( N ). -Step 1 1 − QCF are differentiable a.e. and | df | n = J f . -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 x 1 , ..., x n near every point in N -Step 5 The functions f i = x i ◦ f are n − harmonic in M and |∇ g f i | 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 H f ( 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 L Q u by means of the following identity � � |∇ H u | Q − 2 �∇ H u , ∇ H φ � g d vol , L Q u φ d vol := M M for any φ ∈ W 1 , Q ( M )). If u ∈ W 2 , 2 H , loc ( M ) ∩ W 1 , Q H , loc ( M ) one can 0 then write L Q u = X ∗ i ( |∇ H u | Q − 2 X i u ) 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 � |∇ H u | Q − 2 �∇ H u , ∇ H φ � d vol = 0 , ∀ φ ∈ W 1 , Q H , 0 ( M , vol) . M 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 ( U ) ≤ C . (3) H 2. For any domain K ⊂ M where |∇ H u | 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