Transverse Bochner-Weitzenböck’s formulas and their applications Fabrice Baudoin Geometric Analysis on sub-Riemannian manifolds September 29, 2014 Based on joint works with Bumsik Kim and Jing Wang
Sub-Riemannian manifold with transverse symmetries Let M be a smooth, connected manifold with dimension n + m . We assume that M is equipped with a sub-bundle H ⊂ T M of dimension n and a fiberwise inner product g H on that distribution. ◮ The distribution H is referred to as the set of horizontal directions . ◮ Sub-Riemannian geometry is the study of the geometry which is intrinsically associated to ( H , g H ) .
Riemannian foliations In general, there is no canonical vertical complement of H in the tangent bundle T M , but in many interesting cases H can be seen as the horizontal distribution of a Riemannian foliation F . In this talk, we will assume that the foliation F is totally geodesic with a bundle like metric g . Examples: ◮ The Hopf fibration S 1 → S 2 n + 1 → CP n induces a sub-Riemannian structure on S 2 n + 1 which comes from a totally geodesic foliation. ◮ The quaternionic Hopf fibration SU ( 2 ) → S 4 n + 3 → HP n induces a sub-Riemannian structure on S 4 n + 3 which comes from a totally geodesic foliation. More generally, totally geodesic Riemannian submersions, Sasakian and 3-Sasakian manifolds provide examples of sub-Riemannian structures associated with totally geodesic foliations.
Canonical variation of the metric The metric g can be split as g = g H ⊕ g V , The one-parameter family of Riemannian metrics: g ε = g H ⊕ 1 ε > 0 , ε g V , is called the canonical variation of g . The sub-Riemannian limit is ε → 0. We are interested in a satifying notion of sub-Riemannian Ricci curvature. An easy computation shows that for horizontal vectors X , Y , Ricci ε ( X , Y ) = Ricci H ( X , Y ) − 1 2 ε � JX , JY � . So Ricci ε blows up to −∞ on the horizontal bundle when ε → 0.
Generalized sub-Riemannian Ricci curvature bounds The Lott-Villani-Sturm theory does not apply in the sub-Riemannian framework. Two lines of research: ◮ Eulerian approach : B. and Garofalo (2009) introduce a generalized curvature dimension inequality based on the Bochner’s method (Bakry-Émery Γ 2 - calculus). Later generalizations: B.-Wang (2012), Grong-Thalmaier (2014) ◮ Lagrangian approach: Juillet (2009), Agrachev-Lee (2009) prove a measure contraction property in some sub-Riemannian situations. Later extensions/generalizations have been proposed by several authors including: Barilari, Li, Rifford, Rizzi, Zelenko,... The two approaches have each their advantages and are not yet unified.
Generalized curvature dimension inequality The generalized curvature dimension by B.-Garofalo was originally proved in the context of sub-Riemannian manifolds with transverse symmetries. It has been proved to imply the following results among other things: ◮ Subelliptic Li-Yau estimates, Scale-invariant parabolic Harnack inequalities (B.-Garofalo, to appear JEMS); ◮ Volume doubling property, Poincaré inequality on balls (B.-Bonnefont-Garofalo, Math. Ann. 2012); ◮ Boundedness of the Riesz transform (B.-Garofalo, IMRN 2013).
Generalized curvature dimension inequality The generalized curvature dimension inequality does not give sharp constants in functional inequalities. In a recent work with B. Kim and J. Wang (2014), we prove a transverse Weitzenböck formula in the framework of totally geodesic foliations. As a consequence, the generalized curvature dimension estimate is true in a larger class of examples. It also allows to deduce a sharp lower bound for the first eigenvalue of the sub-Laplacian.
The Bott connection There is a canonical connection on M , the Bott connection, which is given as follows: X Y ) , X , Y ∈ Γ ∞ ( H ) π H ( ∇ R π H ([ X , Y ]) , X ∈ Γ ∞ ( V ) , Y ∈ Γ ∞ ( H ) ∇ X Y = π V ([ X , Y ]) , X ∈ Γ ∞ ( H ) , Y ∈ Γ ∞ ( V ) π V ( ∇ R X Y ) , X , Y ∈ Γ ∞ ( V ) where ∇ R is the Levi-Civita connection and π H (resp. π V ) the projection on H (resp. V ). It is easy to check that for every ε > 0, this connection satisfies ∇ g ε = 0.
The horizontal Laplacian The horizontal Laplacian is the generator of the symmetric Dirichlet form � E H ( f , g ) = �∇ H f , ∇ H g � H d µ. M It is a diffusion operator L on M which is symmetric on C ∞ 0 ( M ) with respect to the volume measure µ . If H is bracket generating, then L is subelliptic. For Z ∈ V , we consider the unique skew-symmetric map J Z defined on the horizontal bundle H such that for every horizontal vector fields X and Y , g H ( J Z ( X ) , Y ) = g V ( Z , T ( X , Y )) .
The transverse Bochner-Weitzenböck formulas Theorem (B., Kim, Wang 2014) Let H ) + 1 H ) ∗ ( ∇ H − T ε ε J ∗ J − Ric H . � ε = − ( ∇ H − T ε Then, for every smooth function f on M , dLf = � ε df , and for any smooth one-form η , 1 Ric H ( η ) − 1 � � ε J ∗ J ( η ) , η 2 L � η � 2 H η � 2 ε −� � ε η, η � ε = �∇ H η − T ε ε + . H
The Bochner-Weitzenböck formulas Where does � ε come from ? The following lemma is easy to prove in an horizontal normal frame. Lemma Let � ∞ = L + 2 J − Ric H . Then, we have for every smooth function f , (1) dLf = � ∞ df . h � J = J Z m ( d ι Z m ) . m = 1
The Bochner-Weitzenböck formulas Since d 2 = 0, if Λ is any fiberwise linear map from the space of two-forms into the space of one-forms, then we have dLf = ( � ∞ + Λ ◦ d ) df . This raises the question of an optimal choice of Λ . Lemma For any Λ and any x ∈ M , we have � 1 � 2 ( L � η � 2 inf ε )( x ) − � ( � ∞ + Λ ◦ d ) η ( x ) , η ( x ) � ε η, � η ( x ) � ε = 1 � 1 � ∞ − 1 �� � � � 2 ( L � η � 2 inf ≤ ε )( x ) − ε T ◦ d η ( x ) , η ( x ) , η, � η ( x ) � ε = 1 ε
The Bochner-Weitzenböck formulas Finally, a new computation in a horizontal normal frame shows that Lemma � ∞ − 2 ε T ◦ d = � ε
Sharp lower bound The Bochner-Weitzenböck formulas have several consequences. Theorem (B.-Kim , 2014) Assume that for every smooth horizontal one-form η , � Ric H ( η ) , η � H ≥ ρ 1 � η � 2 � J ∗ J ( η ) , η � H ≤ κ � η � 2 H , H , and that for every Z ∈ V , Tr ( J ∗ Z J Z ) ≥ ρ 2 � Z � 2 V , with ρ 1 , ρ 2 > 0 and κ ≥ 0 . Then the first eigenvalue λ 1 of the sub-Laplacian − L satisfies ρ 1 λ 1 ≥ . 1 − 1 d + 3 κ ρ 2
Sharp lower bound The bound is sharp: ◮ For the Hopf fibration U ( 1 ) → S 2 d + 1 → CP d , λ 1 = 2 d . On the other hand, for this example, ρ 1 = 2 ( d + 1 ) , κ = 1, ρ 2 = 2 d . ◮ For the quaternionic Hopf fibration SU ( 2 ) → S 4 d + 3 → HP d , λ 1 = d . For this example, ρ 1 = d + 2, κ = 3, ρ 2 = 4 d . Actually we even proved that that for some Riemannian foliations that we called H-type, the equality λ 1 = implies that the ρ 1 1 − 1 n + 3 κ ρ 2 foliation is equivalent to the classical or the quaternionic Hopf fibration.
Curvature dimension inequality Using the Bochner-Weitzenböck formulas, we can also quickly recover the generalized curvature dimension inequality first discovered by B.-Garofalo (2009) in a less general framework by using Γ -calculus If f ∈ C ∞ ( M ) , we denote Γ 2 ( f ) = 1 2 L �∇ H f � 2 − �∇ H f , ∇ H Lf � H and 2 ( f ) = 1 2 L �∇ V f � 2 − �∇ V f , ∇ V Lf � V . Γ Z
Curvature dimension inequality Theorem (B., Kim, Wang 2014) Assume that for every smooth horizontal one-form η , � J ∗ J ( η ) , η � H ≤ κ � η � 2 � Ric H ( η ) , η � H ≥ ρ 1 � η � 2 H , H , and that for every Z ∈ V , Tr ( J ∗ Z J Z ) ≥ ρ 2 � Z � 2 V , with ρ 1 ∈ R , ρ 2 > 0 and κ ≥ 0 . Then for every ν > 0 , 2 ( f ) ≥ 1 ρ 1 − κ �∇ H f � 2 + ρ 2 d ( Lf ) 2 + � � Γ 2 ( f ) + ν Γ Z 4 �∇ V f � 2 ν
Bonnet-Myers theorem As proved in B.-Garofalo, a notable consequence of the generalized curvature dimension inequality is the Bonnet-Myers result. Theorem Assume that for every smooth horizontal one-form η , � Ric H ( η ) , η � H ≥ ρ 1 � η � 2 � J ∗ J ( η ) , η � H ≤ κ � η � 2 H , H , and that for every Z ∈ V , 1 4 Tr ( J ∗ Z J Z ) ≥ ρ 2 � Z � 2 V , with ρ 1 , ρ 2 > 0 and κ ≥ 0 . Then the manifold M is compact and the following diameter bound for the CC distance holds: � √ 1 + 3 κ κ + ρ 2 � � diam ( M ) ≤ 2 3 π n . 2 ρ 2 ρ 1 ρ 2
Bonnet-Myers theorem To put things in perspective, we point out that Ricci ε ( Z , Z ) = Ricci V ( Z , Z ) + 1 4 ε 2 Tr ( J ∗ Z J Z ) Ricci ε ( X , Z ) = 0 Ricci ε ( X , X ) = Ricci H ( X , X ) − 1 2 ε � JX � 2
Volume doubling property, Poincaré inequality on balls Theorem Assume that for every smooth horizontal one-form η , � Ric H ( η ) , η � H ≥ ρ 1 � η � 2 � J ∗ J ( η ) , η � H ≤ κ � η � 2 H , H , and that for every Z ∈ V , Tr ( J ∗ Z J Z ) ≥ ρ 2 � Z � 2 V , with ρ 1 ≥ 0 , ρ 2 > 0 and κ ≥ 0 . Then, there exist constants C d , C p > 0 , for which one has for every x ∈ M and every r > 0 : µ ( B ( x , 2 r )) ≤ C d µ ( B ( x , r )); � � | f − f B | 2 d µ ≤ C p r 2 �∇ H f � 2 d µ, B ( x , r ) B ( x , r ) for every f ∈ C 1 ( B ( x , r )) .
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