Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes On the Hausdorff volume in Sub-Riemannian geometry Davide Barilari (SISSA, Trieste) Nonlinear Control and Singularities, Porquerolles, France October 25, 2010 Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 1 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Joint work with General results and Corank 1 case: Andrei Agrachev Ugo Boscain Corank 2 case: Jean-Paul Gauthier Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 2 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Outline Sub-riemannian geometry 1 Intrinsic volume and sub-Laplacian 2 Comparison between volumes 3 Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 3 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Outline Sub-riemannian geometry 1 Intrinsic volume and sub-Laplacian 2 Comparison between volumes 3 Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 4 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Sub-Riemannian manifolds Definition A sub-Riemannian manifold is a triple S = ( M , ∆ , �· , ·� ) , where ( i ) M is a connected smooth manifold of dimension n ≥ 3; ( 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 . ( iii ) �· , ·� q is a Riemannian metric on ∆ q , that is smooth as function of q . We assume that the Hörmander condition is satisfied Lie q ∆ = T q M , ∀ q ∈ M , where ∆ denotes the set of horizontal vector fields on M , i.e. ∆ = { X ∈ Vec M | X ( q ) ∈ ∆ q , ∀ q ∈ M } . Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 5 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes horizontal curve ∆( q ) A Lipschitz continuous curve γ : [ 0 , T ] → M is said to be horizontal if γ ( t ) ∈ ∆ γ ( t ) ˙ for a.e. t ∈ [ 0 , T ] . Given an horizontal curve γ : [ 0 , T ] → M , the length of γ is � T � ℓ ( γ ) = � ˙ γ, ˙ γ � dt . 0 Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 6 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Carnot-Caratheodory distance The distance induced by the sub-Riemannian structure on M is d ( q , q ′ ) = inf { ℓ ( γ ) | γ ( 0 ) = q , γ ( T ) = q ′ , γ horizontal } . (1) From the Hörmander condition (and connectedness of M ) it follows: d ( q , q ′ ) < ∞ , ∀ q , q ′ ∈ M ( M , d ) is a metric space and d ( · , · ) is continuous with respect to the topology of M (Chow’s Theorem) The function d ( · , · ) is also called Carnot-Caratheodory distance . Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 7 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Orthonormal frame Locally, the pair (∆ , �· , ·� ) can be given by assigning a set of k smooth vector fields, called a local orthonormal frame , spanning ∆ and that are orthonormal ∆ q = span { X 1 ( q ) , . . . , X k ( q ) } , � X i ( q ) , X j ( q ) � = δ ij . The problem of finding curves that minimize the length between two given points q 0 , q 1 , is rewritten as the optimal control problem k � q = ˙ u i X i ( q ) i = 1 � � � T k � � � u 2 i → min 0 i = 1 q ( 0 ) = q 0 , q ( T ) = q 1 Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 8 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Growth vector Define ∆ 1 := ∆ , ∆ 2 := ∆ + [∆ , ∆] , ∆ i + 1 := ∆ i + [∆ i , ∆] . If the dimension of ∆ i q , i = 1 , . . . , m does not depend on the point q ∈ M , the sub-Riemannian manifold is called regular . In the regular case the Hörmander condition guarantees that there exist m ∈ N , such that ∆ m q = T q M , for all q ∈ M . m is called the step of the structure, growth vector of the structure is the sequence , dim ∆ 2 , . . . , dim ∆ m G ( S ) := ( dim ∆ ) � � n k Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 9 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Basic features in SRG t 2 t spheres are highly non-isotropic and they are not smooth even for small time In the regular case the Hausdorff dimension of ( M , d ) is given by the formula m � k i := dim ∆ i − dim ∆ i − 1 . (Mitchell) Q = ik i , i = 1 In particular the Hausdorff dimension is always bigger than the topological dimension of M : Q > n Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 10 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Outline Sub-riemannian geometry 1 Intrinsic volume and sub-Laplacian 2 Comparison between volumes 3 Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 11 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Motivation How to define an intrinsic volume in SRG? (in the sense that it depends only on the sub-Riemannian structure and not on the choice of the coordinates and of the orthonormal frame) This question naturally arise if one wants to study the Heat equation in the sub-Riemannian setting, in order to study the interplay between the analysis of the sub-Laplacian operator L and the geometric structure of M . analysis ← → geometry ? ∂ t φ = L φ ← → curvature What we need first is a “geometric” definition of the Laplacian. Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 12 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Riemannian case On a Riemannian manifold the Laplace-Beltrami operator L is defined as follows L φ = div ( grad φ ) Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 13 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Riemannian case On a Riemannian manifold the Laplace-Beltrami operator L is defined as follows L φ = div ( grad φ ) the gradient of a function φ is the vector field canonically associated to d φ by the metric , i.e. the unique vector field satisfying � grad φ, X � = d φ ( X ) , ∀ X ∈ Vec M . If X 1 , . . . , X n is a local orthonormal frame n � grad φ = X i ( φ ) X i i = 1 Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 13 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Riemannian case On a Riemannian manifold the Laplace-Beltrami operator L is defined as follows L φ = div ( grad φ ) the divergence of a vector field X says how much the flow of X change the volume div X<0 div X>0 If µ is a volume on M , div X is the unique function satisfying L X µ = ( div X ) µ where L X denotes Lie derivative. To define the Laplacian we need to fix a volume! Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 13 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Volume on a Riemannian manifold In the Riemannian case the geometric structure defines the canonical volume: if X 1 , . . . , X n is a local orthonormal frame, the Riemannian volume µ is the volume associated to the n -form � ω µ = dX 1 ∧ . . . ∧ dX n , µ ( A ) = ω µ A With respect to the Riemannian volume we can write n � div X = dX j [ X j , X ] j = 1 and the Laplace-Beltrami operator is written n � X 2 L φ = i φ + ( dX j [ X j , X i ]) X i φ i , j = 1 Example. n = 2 and [ X 1 , X 2 ] = a 1 X 1 + a 2 X 2 L = X 2 1 + X 2 2 − a 2 X 1 + a 1 X 2 Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 14 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Volume on a Riemannian manifold In the Riemannian case the geometric structure defines the canonical volume: if X 1 , . . . , X n is a local orthonormal frame, the Riemannian volume µ is the volume associated to the n -form � ω µ = dX 1 ∧ . . . ∧ dX n , µ ( A ) = ω µ A With respect to the Riemannian volume we can write n � div X = dX j [ X j , X ] j = 1 and the Laplace-Beltrami operator is written n � X 2 L φ = i φ + ( dX j [ X j , X i ]) X i φ i , j = 1 Example. n = 2 and [ X 1 , X 2 ] = a 1 X 1 + a 2 X 2 L = X 2 1 + X 2 2 − a 2 X 1 + a 1 X 2 Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 14 / 37
Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes Hausdorff measure Moreover since ( M , d ) is a metric space we can define the Hausdorff volume H n the spherical Hausdorff volume S n Recall that � � H n (Ω) = lim inf diam ( U i ) n , δ → 0 { Ω ⊂ U i , diam ( U i ) < δ } i i Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 15 / 37
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