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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 ⊂ TM of dimension n and a fiberwise inner product gH 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, gH).

Riemannian foliations

In general, there is no canonical vertical complement of H in the tangent bundle TM, 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 S1 → S2n+1 → CPn induces a

sub-Riemannian structure on S2n+1 which comes from a totally geodesic foliation.

◮ The quaternionic Hopf fibration SU(2) → S4n+3 → HPn

induces a sub-Riemannian structure on S4n+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 = gH ⊕ gV, The one-parameter family of Riemannian metrics: gε = gH ⊕ 1 εgV, ε > 0, 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 , Ricciε(X, Y ) = RicciH(X, Y ) − 1 2εJX, JY . So Ricciε 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 Γ2- 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: ∇XY =            πH(∇R

XY ), X, Y ∈ Γ∞(H)

πH([X, Y ]), X ∈ Γ∞(V), Y ∈ Γ∞(H) πV([X, Y ]), X ∈ Γ∞(H), Y ∈ Γ∞(V) πV(∇R

XY ), X, Y ∈ Γ∞(V)

where ∇R is the Levi-Civita connection and πH (resp. πV) the projection on H (resp. V). It is easy to check that for every ε > 0, this connection satisfies ∇gε = 0.

The horizontal Laplacian

The horizontal Laplacian is the generator of the symmetric Dirichlet form EH(f , g) =

• M

∇Hf , ∇HgHdµ. 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 JZ defined

• n the horizontal bundle H such that for every horizontal vector

fields X and Y , gH(JZ(X), Y ) = gV(Z, T(X, Y )).

The transverse Bochner-Weitzenböck formulas

Theorem (B., Kim, Wang 2014)

Let ε = −(∇H − Tε

H)∗(∇H − Tε H) + 1

εJ∗J − RicH. Then, for every smooth function f on M, dLf = εdf , and for any smooth one-form η, 1 2Lη2

ε−εη, ηε = ∇Hη−Tε Hη2 ε+

• RicH(η) − 1

εJ∗J(η), η

• 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 + 2J − RicH. Then, we have for every smooth function f , dLf = ∞df . (1) J =

h

• m=1

JZm(dιZm).

The Bochner-Weitzenböck formulas

Since d2 = 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 inf

η,η(x)ε=1

1 2(Lη2

ε)(x) − (∞ + Λ ◦ d)η(x), η(x)ε

• ≤

inf

η,η(x)ε=1

1 2(Lη2

ε)(x) −

• ∞ − 1

εT ◦ d

• η(x), η(x)
• ε
• ,

The Bochner-Weitzenböck formulas

Finally, a new computation in a horizontal normal frame shows that

Lemma

∞ − 2 εT ◦ 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, RicH(η), ηH ≥ ρ1η2

H,

J∗J(η), ηH ≤ κη2

H,

and that for every Z ∈ V, Tr(J∗

ZJZ) ≥ ρ2Z2 V,

with ρ1, ρ2 > 0 and κ ≥ 0. Then the first eigenvalue λ1 of the sub-Laplacian −L satisfies λ1 ≥ ρ1 1 − 1

d + 3κ ρ2

.

Sharp lower bound

The bound is sharp:

◮ For the Hopf fibration U(1) → S2d+1 → CPd, λ1 = 2d. On

the other hand, for this example, ρ1 = 2(d + 1), κ = 1, ρ2 = 2d.

◮ For the quaternionic Hopf fibration SU(2) → S4d+3 → HPd,

λ1 = d. For this example, ρ1 = d + 2, κ = 3, ρ2 = 4d. Actually we even proved that that for some Riemannian foliations that we called H-type, the equality λ1 =

ρ1 1− 1

n + 3κ ρ2

implies that the 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 Γ2(f ) = 1 2L∇Hf 2 − ∇Hf , ∇HLf H and ΓZ

2 (f ) = 1

2L∇Vf 2 − ∇Vf , ∇VLf V.

Curvature dimension inequality

Theorem

(B., Kim, Wang 2014) Assume that for every smooth horizontal

• ne-form η,

RicH(η), ηH ≥ ρ1η2

H,

J∗J(η), ηH ≤ κη2

H,

and that for every Z ∈ V, Tr(J∗

ZJZ) ≥ ρ2Z2 V,

with ρ1 ∈ R, ρ2 > 0 and κ ≥ 0. Then for every ν > 0, Γ2(f ) + νΓZ

2 (f ) ≥ 1

d (Lf )2 +

• ρ1 − κ

ν

• ∇Hf 2 + ρ2

4 ∇Vf 2

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, RicH(η), ηH ≥ ρ1η2

H,

J∗J(η), ηH ≤ κη2

H,

and that for every Z ∈ V, 1 4Tr(J∗

ZJZ) ≥ ρ2Z2 V,

with ρ1, ρ2 > 0 and κ ≥ 0. Then the manifold M is compact and the following diameter bound for the CC distance holds: diam(M) ≤ 2 √ 3π

• κ + ρ2

ρ1ρ2

• 1 + 3κ

2ρ2

• n.

Bonnet-Myers theorem

To put things in perspective, we point out that Ricciε(Z, Z) = RicciV(Z, Z) + 1 4ε2 Tr(J∗

ZJZ)

Ricciε(X, Z) = 0 Ricciε(X, X) = RicciH(X, X) − 1 2εJX2

Volume doubling property, Poincaré inequality on balls

Theorem

Assume that for every smooth horizontal one-form η, RicH(η), ηH ≥ ρ1η2

H,

J∗J(η), ηH ≤ κη2

H,

and that for every Z ∈ V, Tr(J∗

ZJZ) ≥ ρ2Z2 V,

with ρ1 ≥ 0, ρ2 > 0 and κ ≥ 0. Then, there exist constants Cd, Cp > 0, for which one has for every x ∈ M and every r > 0: µ(B(x, 2r)) ≤ Cd µ(B(x, r));

• B(x,r)

|f − fB|2dµ ≤ Cpr2

• B(x,r)

∇Hf 2dµ, for every f ∈ C 1(B(x, r)).