The Laplace-Beltrami operator on rank-varying and non-equiregular SR - PowerPoint PPT Presentation

The Laplace-Beltrami operator on rank-varying and non-equiregular SR manifolds (state of the art and new results) Ugo Boscain (CNRS, CMAP, Ecole Polytechnique, Paris) October 12, 2014

Plan of the talk: definition of rank-varying and non-equiregular SR structures definition of the SR Laplacian the choice of an intrinsic volume the singular set acts as a barrier (for the heat and Schroedinger equation) stochastic completeness some related results on conic and anti-conic type surfaces

Classical sub-Riemannian manifolds Definition A classical sub-Riemannian manifold is a triple ( M, � , g ), where ( i ) M is a connected smooth manifold of dimension n ; ( ii ) � is a smooth distribution of constant rank k ≤ n , i.e. a smooth map that associates to q ∈ M a k -dimensional subspace � q of T q M satisfying the LBG condition ( iii ) g q is a Riemannian metric on � q , that is smooth as function of q . → Define � 1 ( q ) := � ( q ), � i +1 := � i + [ � i , � ]. If dim( � i ) , i = 1 , . . . , m do not depend on the point, it is called equiregular otherwise non-equiregular . G =(dim( � 1 ) , dim( � 2 ) , . . . ) is the growth vector . → Locally, ( � , g ) can be given by assigning a set of k smooth vector fields (called a local orthonormal frame ) s.t. � q = span { X 1 ( q ) , . . . , X k ( q ) } , g q ( X i ( q ) , X j ( q )) = δ ij . (1) The Carnot Carath´ eodory distance � � 1 � k k � � � d ( q 1 , q 2 ) := inf { u 2 i ( t ) dt | γ (0) = q 1 , γ (1) = q 2 , ˙ γ ( t ) = u i ( t ) X i ( γ ( t )) } � 0 i =1 i =1

Rank-varying sub-Riemannian manifolds Rank-varying sub-Riemannian structures are structures for which also dim( � )=dim( � 1 ) depends on the point. Locally they are defined by a set of k LBG vector fields that can become collinear, playing the role of an orthonormal frame: → � ( q ) =span { X 1 ( q ) , . . . , X k ( q ) } → The Carnot Carath´ eodory distance � � 1 k k � � � � u 2 d ( q 1 , q 2 ) := inf { i ( t ) dt | γ (0) = q 1 , γ (1) = q 2 , ˙ γ ( t ) = u i ( t ) X i ( γ ( t )) } � 0 i =1 i =1 Remark One cannot say that there exists a positive definite, symmetric bilinear form g such that g q ( X i ( q ) , X j ( q )) = δ ij . Since on the singular set some vector fields could be zero. → Example: the Grushin plane � 1 � 0 � � ( x, y ) ∈ R 2 , X 1 = , X 2 = , 0 x For x = 0 we would have g ( X 2 (0) , X 2 (0)) = 1 with X 2 (0) = 0.

� � A global definition require the use of a fiber bundle : Definition (sub-Riemannian manifold (possibly rank-varying) A ( n, k )- sub-Riemannian structure on M is a triple ( M, U , f ) where: M is a connected smooth manifold of dimension n ; U is an Euclidean bundle with base M and Euclidean fiber U q i.e. U q is a vector space equipped with a scalar product � ., . � q . We assume dim( U q ) = k (the rank of the structure); f : U → T M is a smooth map that is a morphism of vector bundles, i.e. the following diagram is commutative f � T M (2) U ❉ ❉ ❉ ❉ π ❉ π U ❉ ❉ ❉ M and f is linear on fibers. Here π U and π are the canonical projections. � := { f ◦ σ, σ ∈ Γ( U ) } is Lie bracket generated. Here Γ( U ) is the set of smooth sections of U . the map σ �→ f ◦ σ from Γ( U ) to Vec( U ) is injective.

( n, n ) sub-Riemannian structures are called almost-Riemannian structures (generalized Riemannian structures of dimension n having local orthonormal frames made by n vector fields that can become collinear but satisfy the LBG assumption)

Examples Classical equiregular sub-Riemannian structure: Heisenberg � dim( � 1 ( q )) = 2 ,     1 0 ( x, y, z ) ∈ R 3 , X 1 =  , X 2 =  , 0 1   dim( � 2 ( q )) = 3 . − y/ 2 x/ 2 Classical nonequiregular sub-Riemannian structure: Martinet  dim( � 1 ( q )) = 2 ,     1 0   dim( � 2 ( q )) = 3 . if y � = 0 �  ( x, y, z ) ∈ R 3 , X 1 =  , X 2 =  , 0 1   dim( � 2 ( q )) = 2 . if y = 0 y 2 / 2 0   dim( � 3 ( q )) = 3 .  Rank-varying sub-Riemannian structure: Grushin structure (Baouendi, ’67, Grushin ’70, Franchi-Lanconelli ’84) (almost-Riemannian structure) � 1 � 0  dim( � 1 ( q )) = 2 . if x � = 0 � � �  ( x, y ) ∈ R 2 , X 1 = dim( � 1 ( q )) = 1 . if x = 0 , X 2 = , 0 x dim( � 2 ( q )) = 2 . 

Singular set where some of the � i change dimension

Hypoelliptic diffusion Historically ( Jerison, Fefferman, Sanchez-Calle Hormander, etc.... ) hypoelliptic diffusion has been introduced as the diffusion generated by an operator L second order (without zero order) L = � n i =1 g ij ( q ) ∂ i ∂ j + � n i =1 b i ( q ) ∂ i the matrix g ij is positive semidefinite. a “Lie-Bracket generating condition” is verified. For instance g ij ( q ) = � k h ( q ) B j h =1 B i h ( q ) with B 1 ( q ) , . . . , B k ( q ) smooth vector fields that are Lie-Bracket generated L is sub-elliptic From such an operator one can extract a SR metric as the “inverse” of g ij . Typical examples are the operators in Hormander form k � X 2 L = i + X 0 with X 1 , . . . X k LBG vector fields i =1

Riemannian diffusion This is not the approach that people have in Riemannian geometry where: one starts from the geometric structure ( M, g ) and then extracts the diffusion operator (the Laplace-Beltrami operator, that in this case is elliptic) as ∆( φ ) = div r (grad( φ )) , where the divergence is computed with respect to the Riemannian volume r . This approach has the following advantage: Theorem (Strichartz) Let ( M, g ) be a complete Riemannian manifold. Then the Laplace-Beltrami operator defined on C ∞ 0 ( M ) is essentially self-adjoint in L 2 ( M, r ) and hence it is in one to one correspondence with a strongly continuous semigroup e tL and it admits a smooth symmetric positive L 2 ( M, r ) heat kernel. → thanks to self-adjointness one can also study the Schroedinger equation n √ g ∂ i ( √ gg ij ∂ j φ ) = 1 � X 2 � � ∆( φ ) = i + div r ( X i ) X i , i =1 where { X i } is an orthonormal frame

A result of this kind is less evident starting from a diffusion operator, because one does not know a priori a volume w.r.t. which guarantee self-adjointness, unless: one starts from an operator in divergence form, one adjusts the first order terms with the volume, one renounce to a good evolution in L 2 (and to consider the Schroedinger equation) Remark We say that in Riemannian geometry the Laplace-Beltrami operator is intrinsic since it depends only on the Riemannian structure and not choice of coordinates choice of the orthonormal frame choice of the volume (the volume is fixed by the Riemannian structure)

The intrinsic Laplacian in SRG The most natural construction for a diffusion operator in SR geometry is to follow the Riemannian construction and to define: ∆ H ( φ ) = div(grad H ( φ )) the horizontal gradient is grad( φ ) = � k i =1 X i ( φ ) X i the divergence is the standard divergence computed with respect to an “intrinsic” volume that we have to determine This approach has the following advantages: In the equiregular case if the sub-Riemannian manifold is complete as metric space we have self-adjointness and existence and uniqueness of a strongly continuous semigroup (Strichartz). In the rank-varying or nonequiregular case we will see appearing some interesting phenomena in the heat diffusion, deeply connecting to the geometry of the SRM.

Remarks Remark 1 While in the equiregular case the choice of the volume does not affect certain properties as small time heat kernel asymptotics Weyl law in the rank-varying case or in the non equiregular case the choice of an intrinsic volume will crucially affect these properties. Remark 2 The expression of the intrinsic Laplacian is k � X 2 � � ∆ H = i + div( X i ) X i , i =1 where { X i } is an orthonormal frameof the SRM Notice that it is not the sum of squares if we start from a diffusion operator L , we extract the metric and we build the intrinsic Laplacian, we will not obtain in general L . diffusion operator L − → sub-Riemannian structure − → ∆ H � = L (in the rank-varying or non equir. cases they can be drastically different)

Does an intrinsic volume exist in SRG? An “intrinsic volume” is a volume that is constructed only from the sub-Riemannian structure and that does not depend on choice of the coordinates choice of the orthonormal frame In the equiregular case there are several constructions of intrinsic volumes: Popp’s volume P (Montgomery’s book) “algebraic”. It is always C ∞ Spherical Q -Hausdorff volume S Q - Hausdorff volume H others ..... Relations between these volumes: → It is known that they are commensurable (Mitchell): The Radon-Nikodym derivatives are measurable functions that are locally bounded and locally bounded away from zero → for left-invariant sub-Riemannian structures on Lie groups they are proportional → in the Riemannian case they are proportional → Montgomery question: are they proportional in general?