BISHOP AND LAPLACIAN COMPARISON THEOREMS ON …BISHOP AND LAPLACIAN COMPARISON THEOREMS ON THREE DIMENSIONAL CONTACT SUBRIEMANNIAN MANIFOLDS WITH SYMMETRY ANDREI AGRACHEV AND PAUL

BISHOP AND LAPLACIAN COMPARISON THEOREMSON THREE DIMENSIONAL CONTACT

SUBRIEMANNIAN MANIFOLDS WITH SYMMETRY

ANDREI AGRACHEV AND PAUL W.Y. LEE

Abstract. We prove a Bishop volume comparison theorem and aLaplacian comparison theorem for three dimensional contact sub-riemannian manifolds with symmetry.

1. Introduction

Recently, there are numerous progress in the understanding of curva-ture type invariants in subriemannian geometry and their applicationsto PDE [10, 12, 13, 1, 2, 3, 4, 14]. In this paper, we continue to in-vestigate some consequences on bounds of these curvature invariants.More precisely, we prove a Bishop comparison theorem and a Lapla-cian comparison theorem for three dimensional contact subriemannianmanifolds with symmetry (also called Sasakian manifolds).

The paper is organized as follows. In section 2, we recall variousnotions in subriemannian geometry needed in this paper. In particular,we recall the definition of curvature R11 and R22 for three dimensionalcontact subriemannian manifolds introduced in [12, 13, 1]. In section3, we show that the curvature R11 is closely related to the Tanaka-Webster curvature in CR geometry. In section 4, we collect variousresults on the cut loci of Sasakian manifold with constant Tanaka-Webster curvature (also called Sasakian space forms). In section 5, wegive an estimate for the volume of subriemannian balls. In section 6, weprove the subriemannian Bishop theorem which compares the volumeof subriemannian balls of a Sasakian manifold and a Sasakian spaceform. We introduce the subriemannian Hessian and sub-Laplacian insection 7 and give the formula for the Laplacian of the subriemanniandistance in Sasakian space form in section 8. We prove a subriemannianHessian and a subriemannian Laplacian comparison theorem in section

Date: May 11, 2011.The first author was partially supported by the PRIN project and the second

author was supported by the NSERC postdoctoral fellowship.1

2 ANDREI AGRACHEV AND PAUL W.Y. LEE

9. As an application, we give a lower bound of the solution to thesubriemannian heat equation in section 10.

Acknowledgment

The authors would like to thank N. Garofalo for the stimulatingdiscussions.

2. Subriemannian Geometry

In this section, we recall various notions in subriemannian geometryneeded in this paper. A subriemannian manifold is a triple (M,Δ, g),where M is a smooth manifold, Δ is a distribution (a vector subbundleof the tangent bundle of M), and g is a fibrewise inner product definedon the distribution Δ. The inner product g is also called a subrieman-nian metric. An absolutely continuous curve : [0, 1] → M on themanifold M is called horizontal if it is almost everywhere tangent tothe distribution Δ. The distribution Δ is called bracket-generating ifvector fields contained in Δ together with their iterated Lie bracketsspan the whole tangent bundle. More precisely, let Δ1 and Δ2 be twodistributions on a manifold M , and let X(Δi) be the space of all vectorfields contained in the distribution Δi. Let [Δ1,Δ2] be the distributiondefined by

[Δ1,Δ2]x = span{w1(x), [w2, w3](x)∣wi ∈ X(Δi)}.We define inductively the following distributions: [Δ,Δ] = Δ2 and

Δk = [Δ,Δk−1]. A distribution Δ is called bracket generating if Δk =TM for some k. Under the bracket generating assumption, we havethe following famous Chow-Rashevskii Theorem (see [15] for a proof):

Theorem 2.1. (Chow-Rashevskii) Assume that the manifold M is con-nected and the distribution Δ is bracket generating, then there is ahorizontal curve joining any two given points.

Assuming the distribution Δ is bracket generating, we can use theinner product g to define the length l( ) of a horizontal curve by

l( ) =

∫ 1

0

g( (t), (t))1/2dt.

The subriemannian or Carnot-Caratheodory distance d(x, y) betweentwo points x and y on the manifold M is defined by

(2.1) d(x, y) = inf l( ),

where the infimum is taken over all horizontal curves which start fromx and end at y.

COMPARISON THEOREMS ON CONTACT SUBRIEMANNIAN MANIFOLDS 3

The horizontal curves which realize the infimum in (2.1) are calledlength minimizing geodesics. From now on all manifolds are assumedto be complete with respect to a given subriemannian distance. Itmeans that given any two points on the manifold, there is at least onegeodesic joining them.

Next we discuss the geodesic equation in the subriemannian setting.Let � be a covector in the cotangent space T ∗xM at the point x. Bynondegeneracy of the metric g, we can define a vector v in the distribu-tion Δx such that g(v, ⋅) coincides with �(⋅) on Δx. The subriemannianHamiltonian H corresponding to the subriemannian metric g is definedby

H(�) :=1

2g(v, v).

Note that this construction defines the usual kinetic energy Hamilton-ian in the Riemannian case.

Let � : T ∗M → M be the projection map. The tautological oneform � on T ∗M is defined by

��(V ) = �(d�(V )),

where � is in the cotangent bundle T ∗M and V is a tangent vector onthe manifold T ∗M at �.

Let ! = d� be the symplectic two form on T ∗M . The Hamilton-ian vector field H corresponding to the Hamiltonian H is defined by!(H, ⋅) = −dH(⋅). By the nondegeneracy of the symplectic form !, the

Hamiltonian vector field H is uniquely defined. We denote the flow cor-

responding to the vector field H by etH . If t 7→ etH(�) is a trajectory of

the above Hamiltonian flow, then its projection t 7→ (t) = �(etH(�))is a locally minimizing geodesic. That means sufficiently short segmentof the curve is a minimizing geodesic between its endpoints. The min-imizing geodesics obtained this way are called normal geodesics. In thespecial case where the distribution Δ is the whole tangent bundle TM ,the distance function (2.1) is the usual Riemannian distance and allgeodesics are normal. However, this is not the case for subriemannianmanifolds in general (see [15] and reference therein for more detail).

Next we restrict our attension to the three dimensional contact sub-riemannian manifold. Let Δ be a two generating distribution with twodimensional fibres on a three dimensional manifold M . Δ is a contactdistribution if there exists a covvector � such that Δ = {v∣�(v) = 0}and the restriction of d� to Δ is nondegenerate. If we fix a subrieman-nian metric g, then we can choose � so that the restriction of d� tothe distribution Δ coincides with the volume form with respect to thesubriemannian metric g.


Let {v1, v2} be a local orthonormal frame in the distribution Δ withrespect to the subriemannian metric g and let v0 be the Reeb fielddefined by the conditions �(v0) = 1 and d�(v0, ⋅) = 0. This defines aframe {v0, v1, v2} in the tangent bundle TM and we let {�0 = �, �1, �2}be the corresponding dual co-frame in the cotangent bundle T ∗M (i.e.�i(vj) = �ij).

The frame {v0, v1, v2} and the co-frame {�0, �1, �2} defined aboveinduces a frame in the tangent bundle TT ∗M of the cotangent bundleT ∗M . Indeed, let �i be the vector fields on the cotangent bundle T ∗Mdefined by i�i! = −�i. Note that the symbol �i in the definition of �irepresents the pull back �∗�i of the 1-form � on the manifold M by theprojection � : T ∗M → M . This convention of identifying forms in themanifold M and its pull back on the cotangent bundle T ∗M will be usedfor the rest of this paper without mentioning. Let �1 and �2 be the 1-forms defined by �1 = ℎ1�2 − ℎ2�1 and �2 = ℎ1�1 + ℎ2�2, respectively,

and let �i be the vector fields defined by i�i! = −�i. Finally if welet ℎi : T ∗M → ℝ be the Hamiltonian lift of the vector fields vi,

defined by ℎi(�) = �(vi), then the vector fields ℎ0, ℎ1, ℎ2, �, �1, �2 definea local frame for the tangent bundle TT ∗M of the cotangent bundleT ∗M . Under the above notation the subriemannian Hamiltonian isgiven by H = 1

2((ℎ1)2 + (ℎ2)2) and the Hamiltonian vector field is

H = ℎ1ℎ1 + ℎ2ℎ2.We also need the bracket relations of the vector fields v0, v1, v2. Let

akij be the functions on the manifold M defined by

(2.2) [vi, vj] = a0ijv0 + a1

ijv1 + a2ijv2.

It is not hard to check that

(2.3) a001 = a0

02 = 0, a012 = −1, a1

01 + a202 = 0.

Recall that a basis {e1, ..., en, f1, ..., fn} in a symplectic vector spacewith a symplectic form ! is a Darboux basis if it satisfies !(ei, ej) =!(fi, fj) = 0, and !(fi, ej) = �ij. We recall the following theorem from[1].

Theorem 2.2. For each fixed � in the manifold T ∗M , there is a movingDarboux frame

ei(t) = (etH)∗ei(0), fi(t) = (etH)∗fi(0), i = 1, 2, 3

in the symplectic vector space T�T∗M and functions

R11t = (etH)∗R11

0 , R22t = (etH)∗R22

0 : T ∗M → ℝ


depending on time t such that the following structural equations aresatisfied ⎧⎨⎩

e1(t) = f1(t),e2(t) = e1(t),e3(t) = f3(t),

f1(t) = −R11t e1(t)− f2(t),

f2(t) = −R22t e2(t),

f3(t) = 0.

Moreover,⎧⎨⎩

e1(0) = 1√2H�1,

e2(0) = 1√2H�,

e3(0) = − 1√2H

(ℎ0�0 + ℎ1�1 + ℎ2�2),

f1(0) = 1√2H

[ℎ1ℎ2 − ℎ2ℎ1 + �0�0 + (�1ℎ12)�1 − ℎ12�2],

f2(0) = 1√2H

[2Hℎ0 − ℎ0H − �1�0 + (�1a)�1 − a�2],

f3(0) = − 1√2HH,

R110 = ℎ2

0 + 2H�− 32�1a,

R220 = R11

0 �1a− 3H�1Ha+ 3H2�1a+ �1H2a.

where

a = dℎ0(H),

�0 = ℎ2ℎ01 − ℎ1ℎ02 + �1a,

�1 = ℎ0a+ 2H�1a− �1Ha,� = v1a

212 − v2a

112 − (a1

12)2 − (a212)2 − 1

2(a2

01 − a102).

3. Connection with Tanaka-Webster Scalar Curvature

In this section, we show that the invariant � defined in Theorem2.2 is up to a constant the Tanaka-Webster scalar curvature in CRgeometry.

Let gR be the Riemannian metric on the manifold M such that thebasis v0, v1, v2 is orthonormal. Let K(v, w) be the sectional curvatureof the plane spanned by v and w and let ℜc(v) be the Ricci curvatureof the vector v.

Theorem 3.1. The invariant � satisfies

� = 2K(v1, v2) + ℜc(v0) + 1.

Remark 3.2. From now on, we called � the Tanaka-Webster scalarcurvature. It follows from the above theorem that 4� coincides withthe definition of Tanaka-Webster scalar curvature in [17].


Proof. Let ∇ denotes the Riemannian connection of the above Rie-mannian metric. By Koszul’s formula, we have the following.

∇v1v1 = a101v0 − a1

12v2,

∇v2v2 = a202v0 − a2

21v1,

∇v1v2 =1

2(a1

02 + a201 − 1)v0 + a1

12v1,

∇v2v1 =1

2(a2

01 + a102 + 1)v0 − a2

12v2,

∇v1v0 = −a101v1 +

1

2(−a1

02 − a201 + 1)v2,

∇v2v0 = −a202v2 +

1

2(−a2

01 − a102 − 1)v1,

∇v0v1 = a201v2 +

1

2(−a1

02 − a201 + 1)v2,

∇v0v2 = a102v1 +

1

2(−a2

01 − a102 − 1)v1.

Let ℜ be the Riemann curvature tensor of the Riemannian metricgR. By definition of ℜ, we also have

gR(ℜ(vi, vj)vi, vj)

= gR(∇[vi,vj ]vi −∇vi∇vjvi +∇vj∇vivi, vj).

It follows from the above that

ℜc(v0) = gR(ℜ(v0, v1)v0, v1) + gR(ℜ(v0, v2)v0, v2)

= −2(a101)2 − 1

2(a2

01 + a102)2 +

1

2

and

K(v1, v2) = gR(ℜ(v1, v2)v1, v2)

=1

2(a1

02 − a201)− (a1

12)2 − (a212)2 + v1a

212

− v2a112 + (a1

01)2 +1

4(a1

02 + a201)2 − 3

4.

□


4. Sasakian Space Form

A three dimensional contact subriemannian manifold is Sasakian ifthe Reeb field preserves the subriemannian metric. Using the nota-tion of this paper, it is the same as a = dℎ0(H) = 0. A three di-mensional Sasakian manifold is a Sasakian space form if the Tanaka-Webster scalar curvature is constant. In this section, we collect variousfacts about injectivity domain (see below for the definition) of Sasakianspace forms including the recent results in [6].

Let (M,Δ, g) be a subriemannian manifold. Let H be the sub-

riemannian Hamiltonian and let etH be the Hamiltonian flow. Let� : T ∗M → M be the projection map and let us fix a point x in themanifold M . Let Ωx be the set of all covectors � in the cotangent space

T ∗xM such that the curve : [0, 1] → M defined by (t) = �(etH(�))is a length minimizing geodesic. We call Ω =

∪x Ωx the injectivity

domain of the subriemannian manifold. We also let ΩRx be the set of

covectors in Ωx such that the corresponding curve has length less than

or equal to R. A point � in T ∗xM is a cut point if (t) = �(etH(�)) isminimizing geodesic on [0, 1] and not minimizing on any larger interval.

A cut point � is a conjugate point if the map �(e1⋅H) is singular at �.The Heisenberg group ℍ is a well-known example of Sasakian man-

ifold with vanishing Tanaka-Webster curvature. The manifold in thiscase is given by ℝ3 and the distribution Δ is the span of two vectorfields ∂x − 1

2y∂z and ∂y + 1

2x∂z. This two vector fields also define a

subriemannian metric for which they are orthnormal. In this case allcut points are conjugate points and ΩR is given by

(4.1) ΩR = {�∣√

2H(�) ≤ R,−2� ≤ ℎ0(�) ≤ 2�}.Recall that SU(2), the special unitary group, consists of 2×2 matri-

ces with complex coefficients and determinant 1. The Lie algebra su(2)consists of skew Hermitian matrices with trace zero. The left invariantvector fields of the following two elements in su(2)

u1 =

(0 1/2−1/2 0

), u2 =

(0 i/2i/2 0

)span the standard distribution Δ on SU(2). Let gc be the subrieman-nian metric for which gc(cu1, cu2) = 1. The Reeb field in this case isc2u0, where

u0 =

(0 −1/2

1/2 0

).

A computation shows that the Tanaka-Webster curvature is given byc2. It follows from the result in [6] that all cut points are conjugate


points in this case and ΩR is given by

(4.2) ΩR = {�∣√ℎ0(�)2 + 2c2H(�) ≤ 2�}.

The special linear group SL(2) is the set of all 2 × 2 matrices withreal coefficients and determinant 1. The Lie algebra sl(2) is the set ofall 2× 2 real matrices with trace zero. The left invariant vector fieldsof the following two elements in sl(2)

u1 =

(1/2 00 −1/2

), u2 =

(0 1/2

1/2 0

)span the standard distribution Δ on SL(2). Let gc be the subrieman-nian metric for which gc(cu1, cu2) = 1. The Reeb field in this case isc2u0, where

u0 =

(0 −1/2

1/2 0

).

The Tanaka-Webster curvature is given by −c2. The structure of theset of cut points in this case is much more complicated. However, theresult in [6] and a computation shows the following.

Theorem 4.1. Assume that a cut point � in the cotangent bundle ofSL(2) with subriemannian metric gc is contained in ΩR, where R =2√

2�c

. Then it is a conjugate point. Moreover, it satisfies√∣ℎ0(�)2 − 2c2H(�)∣ = 2�

Proof. Let � = ℎ20 − 2Hc2. From the proof of [6, Theorem 5], � in the

cotangent space T ∗xSL(2) at a point x is a cut point if it satisfies

(4.3)tan(ℎ0(�)/2)

ℎ0(�)=

tanh(√−�(�)/2)√−�(�)

for �(�) < 0,

(4.4)tan(ℎ0(�)/2)

ℎ0(�)=

tan(√�(�)/2)√�(�)

for �(�) > 0, or

(4.5)tan(ℎ0(�)/2)

ℎ0(�)=

1

2

for �(�) = 0.Let r1, r2, r3 be the infimum of 2H(�)c2 where � runs over positive

solutions of (4.3), (4.4),(4.5), respectively. The goal is to find the min-imum of {r1, r2, r3}.


Let f(x) = tan(√x)√

xand g(x) = tanh(

√x)√

x. Let F1 be a branch of inverses

of f∣∣∣[0,∞)

and let G be the inverse of g∣∣∣[0,∞)

. Finding r1 is the same as

minimizing 4(F1 +G).A computation shows that that the derivatives of F1 and G satisfy

(4.6) F ′1(x) =1 + x2F1(x)− x

2F1(x), G′(x) =

1− x2G(x)− x2G(x)

.

Since F1 and G are nonnegative, F ′1 + G′ = 0 implies that x = 1. Itfollows that r1 ≥ r3.

Let F2 be another branch of inverses of f∣∣∣[0,∞)

for which F2 > F1.

We can assume that F1 is the smalllest branch and F2 is the secondsmallest branch. In this case, finding r2 is the same as minimizing4(F2−F1). It follows from (4.6) that F ′2(x)−F ′1(x) = 0 implies x = 1.Therefore, there are two possibilities. Either the minimum of F2 − F1

occurs at x = 1 which implies that r2 ≥ r3 or 4(F2 − F1) goes to the

infimum as x→∞ which implies that r2 = 4(

3�2

)2−(�2

)2= 8�2 < r3.

The last assertion follows from [6]. □

5. Volume of Subriemannian Balls

In this section, we give an estimate on the volume of subriemannianballs of Sasakian manifolds assuming the Tanaka-Webster curvature isbounded below. More precisely, let us fix a point x in the manifoldand let v1, v2 be an orthonormal basis of the subriemannian metricaround x. Let v0 be the Reeb field and �0, �1, �2 be the dual coframeof the frame v0, v1, v2 (i.e. �i(vj) = �ij). We use this coframe to in-troduce coordinates on the cotangent space T ∗xM and let m be thecorresponding Lebesgue measure. Let (r, �, ℎ) be the cylindrical co-ordinates on T ∗xM corresponding to the above coordinate system (i.e.

ℎ = ℎ0(�), r2 = 2H(�), and tan(�) = ℎ2(�)ℎ1(�)

). Recall that ΩR de-

notes the set of all covectors � such that√

2H(�) ≤ R and the curve

t 7→ �(etH(�)), 0 ≤ t ≤ 1 is length minimizing. We use the coordinatesystem introduced above on T ∗xM to identify the set ΩR with a sub-set in ℝn. Finally, let � be the volume form defined by the condition�(v0, v1, v2) = 1. We denote the measure induced by � using the samesymbol.

Theorem 5.1. Assume that there exists a constant k1 (resp. k2) suchthat the Tanaka-Webster curvature � of a three dimensional Sasakian


manifold satisfies � ≥ k1 (resp. ≤ k2) on the ball B(x,R) of radius Rcentered at the point x. Then

�(B(x,R)) ≤∫

ΩR

bk1dm

(resp. ≥

∫ΩR

bk2dm

),

where bk : T ∗xM → ℝ is defined via the above mentioned cylindricalcoordinates by

bk =

⎧⎨⎩r2(2−2 cos(�)−� sin(�))

�2 if � > 0,r2(2−2 cosh(�)+� sinh(�)))

�2 if � < 0,r2

12if � = 0,

� = ℎ2 + r2k, and � =√∣�∣.

As a corollary, we have a formula for the volume of subriemannianballs on Sasakian space forms. Remark that explicit formula for the setΩR in various examples are present in Section 4 (see also [6] for moredetails).

Corollary 5.2. Assume that the three dimensional subriemannian man-ifold is a Sasakian space form with Tanaka-Webster curvature k. Then

�(B(x,R)) =

∫ΩR

bkdm.

Proof of Theorem 5.1. Recall that � is the measure on M defined by

�(v0, v1, v2) = 1. Let t : T ∗xM → M be the map t(�) = �(e1⋅H(�))and let �t : T ∗xM → ℝ be the function defined by ∗t � = �tm. Let usfix a covector � in T ∗xM . Let e1(t), e2(t), e3(t), f1(t), f2(t), f3(t) be acanonical Darboux frame at � defined by Theorem 2.2. Let aij(t) andbij(t) be defined by

(5.1) ei(0) =3∑j=1

(aij(t)ej(t) + bij(t)fj(t)).

Finally let At and Bt be the matrices with (i, j)-th entry equal to aij(t)and bij(t), respectively.

A computation using (5.1) and Theorem 2.2 gives

(5.2) �t = 2H(�) detBt.

It follows that from (5.2) that

(5.3) �(B(x, r)) =

∫ 1(Ωr)

d� =

∫Ωr

∣�1∣dm =

∫Ωr

2H(�)∣ detB1∣dm.


Let Et = (e1(t), e2(t), e3(t))T and let Ft = (f1(t), f2(t), f3(t))T . Herethe superscript T denote matrix transpose. By the definition of thematrices At and Bt, we have

E0 = AtEt +BtFt.

If we differentiate the above equation with respect to time t, we have

0 = AtEt + AtEt + BtFt +BtFt

= AtEt + AtC1Et + AtC2Ft + BtFt +Bt(−RtEt − CT1 Ft),

Since the manifold is Sasakian, a = dℎ0(H) = 0. Therefore, byTheorem 2.2, Et and Ft satisfy the following equatoins

Et = C1Et + C2Ft, Ft = −RtEt − CT1 Ft,

where

C1 =

⎛⎝ 0 0 01 0 00 0 0

⎞⎠ , C2 =

⎛⎝ 1 0 00 0 00 0 1

⎞⎠ ,

Rt =

⎛⎝ R11t 0 00 0 00 0 0

⎞⎠ =

⎛⎝ ℎ20(�) + 2�tH(�) 0 0

0 0 00 0 0

⎞⎠ ,

and �t = �( t(�)).It follows that the matrices At and Bt satisfy the following equations

(5.4) At + AtC1 −BtRt = 0, Bt + AtC2 −BtCT1 = 0

with initial conditions B0 = 0 and A0 = Id.If we set St = B−1

t At and Ut = S−1t = A−1

t Bt, then they satisfy thefollowing Riccati equations.

St − StC2St + CT1 St + SC1 −Rt = 0

and

Ut + UtRtUt − C1Ut − UtCT1 + C2 = 0

with initial condition U0 = 0.Let us fix a constant k and consider the following Riccati equation

with constant coefficients

(5.5) Ukt + Uk

t RkUk

t − C1Ukt − Uk

t CT1 + C2 = 0

and initial condition Uk0 = 0, where Rk =

⎛⎝ ℎ20(�) + 2kH(�) 0 0

0 0 00 0 0

⎞⎠.


The solution of (5.5) can be found by the method in [11]. If ℎ20 +

2H(�)k > 0, then

Ukt =

⎛⎜⎝−t sin(�t)�t cos(�t)

t2(cos(�t)−1)

�2t cos(�t)0

t2(cos(�t)−1)

�2t cos(�t)

t3(�t cos(�t)−sin(�t))

�3t cos(�t)0

0 0 −t

⎞⎟⎠ .

If ℎ20 + 2H(�)k < 0, then

Ukt =

⎛⎜⎝−t sinh(�t)�t cosh(�t)

t2(1−cosh(�t))

�2t cosh(�t)0

t2(1−cosh(�t))

�2t cosh(�t)

t3(sinh(�t)−�t cosh(�t))

�3t cosh(�t)0

0 0 −t

⎞⎟⎠ .

If ℎ20 + 2H(�)k = 0, then

Ukt =

⎛⎝ −t − t2

20

− t2

2− t3

30

0 0 −t

⎞⎠ ,

where �t = t√∣ℎ2

0 + 2H(�)k∣.If we call the inverse Skt = (Uk

t )−1, then

Skt =

⎛⎜⎝�t(�t cos(�t)−sin(�t))

t(2−2 cos(�t)−�t sin(�t))

�2t (1−cos(�t))

t2(2−2 cos(�t)−�t sin(�t))0

�2t (1−cos(�t))

t2(2−2 cos(�t)−�t sin(�t))

−�3t sin(�t)

t3(2−2 cos(�t)−�t sin(�t))0

0 0 −1t

⎞⎟⎠ ,

if ℎ20 + 2H(�)k > 0.

Skt =

⎛⎜⎝�t(sinh(�t)−�t cosh(�t))

t(2−2 cosh(�t)+�t sinh(�t))

�2t (cosh(�t)−1)

t2(2−2 cosh(�t)+�t sinh(�t))0

�2t (cosh(�t)−1)

t2(2−2 cosh(�t)+�t sinh(�t))

−�3t sinh(�t)

t3(2−2 cosh(�t)+�t sinh(�t))0

0 0 −1t

⎞⎟⎠ ,

if ℎ20 + 2H(�)k < 0.

Skt =

⎛⎝ −4t

6t2

06t2−12

t30

0 0 −1t

⎞⎠ ,

if ℎ20 + 2H(�)k = 0.

By [16], if k1 ≤ � ≤ k2, then

(5.6) Uk2t ≤ Ut ≤ Uk1

t ≤ 0.

Therefore, Sk1t ≤ St ≤ Sk2t .


On the other hand, by (5.4) and the definition of St, we have

Bt +Bt(StC2 − CT1 ) = 0.

It follows that ddt

detBt = tr(CT1 − StC2) detBt = −tr(StC2) detBt.

If we replace the matrix Rt in (5.4) by Rk and denote the solution byAkt and Bk

t , then we have

ddt

detBt

detBt

= −tr(StC2) ≥ −tr(Sk2t C2) =ddt

detBk2t

detBk2t

.

It follows that detBt

detBk2t

is nondecreasing. By definition of Ut and (5.6),

we also have

limt→0

detBt

detBk2t

= limt→0

detAt detUt

detAk2t detUk2t

≥ limt→0

detAt

detAk2t= 1

Therefore, it follows that

detBt

detBk2t

≥ limt→0

detBt

detBk2t

≥ 1.

Similarly, we also have

detBt

detBk1t

≤ 1.

A calculation gives

(5.7) ∣ detBkt ∣ := bkt =

t(2− 2 cos(�t)− �t sin(�t))

� 41

if ℎ20 + 2H(�)k > 0,

(5.8) bkt =t(2− 2 cosh(�t) + �t sinh(�t))

� 41

if ℎ20 + 2H(�)k < 0, and

(5.9) bkt =t5

12

if ℎ20 + 2H(�)k = 0.

It follows that bk2t ≤ ∣ detBt∣ ≤ bk1t . Therefore, we have the followingas claimed ∫

ΩR

r2bk21 dm ≤ �(B(x, r)) ≤∫

ΩR

r2bk11 dm.

□


6. Subriemannian Bishop Theorem

In this section, we prove a subriemannian analog of Bishop theoremfor three dimensional Sasakian manifolds. Recall that � is the volumeform defined by the condition �(v0, v1, v2). We denote the measureinduced by � using the same symbol and let �k be the correspondingmeasure in a Sasakian space form of curvature k. Let Bk(R) be asubriemannian ball of radius R in one of the Sasakian space formsSU(2), ℍ, or SL(2) of curvature k (see Section 4 for a discussion ofthese space forms).

Theorem 6.1. (Subriemannian Bishop Theorem) Assume that theTanaka-Webster scalar curvature � of a three dimensional Sasakianmanifold satisfies � ≥ k on the ball B(x,R) for some constant k. Ifk ≥ 0, then

(6.1) �(B(x,R)) ≤ �k(Bk(R))

and equality holds only if � = k on B(x,R). The same conclusion holds

for k < 0 provided that R ≤ 2√

2�c

.

Proof. Let us start with the proof of (6.1). By (4.1), (4.2), Theorem4.1, and Corollary 5.2, it is enough to show that ΩR is contained in theset

{� ∈ T ∗xM ∣√∣ℎ0(�)2 + kH(�)∣ ≤ 2�}

Suppose that there is a covector � in ΩR such that

�1 :=√∣ℎ0(�)2 + kH(�)∣ > 2�.

Using the notations in the proof of Theorem 5.1, we let

E0 = AtEt +BtFt,

where Et = (e0(t), e1(t), e2(t))T and Ft = (f0(t), f1(t), f2(t))T are canon-ical Darboux frame at �.

By the proof of Theorem 5.1, we have that ∣ detBt∣ ≤ bkt , where bkt isdefined in (5.7), (5.8), and (5.9). Since �1 > 2�, it follows that detBt =0 for some t < 1. Therefore, t� is a conjugate point contradicting thefact that � is contained in ΩR.

Next, suppose equality holds in (6.1) and � > k on an open setO contained in the ball B(x,R). For each point y in O, let (t) =

�(etH(�)) be a minimizing geodesic connecting x and y. It follows thatRt > Rk

t for all t close enough to 1. By the result in [16] and a similarargument as in Theorem 5.1, we have ∣ detB1∣ < ∣ detBk

1 ∣. It followsfrom (5.3) that �(B(x,R)) < �k(Bk(R)) which is a contradiction. □


7. Subriemannian Hessian and Laplacian

In this section, we introduce subriemannian versions of Hessian andLaplacian. For the computation, we will also give an expression for itin the canonical Darboux frame.

Assume that a functon f : M → ℝ is twice differentiable at a pointx in the manifold M . The canonical Darboux frame

{e1(t), e2(t), e3(t), f1(t), f2(t), f3(t)}

at dfx gives a splitting of the tangent space TdfxT∗M = ℋ⊕V defined

by

ℋ = span{f1(0), f2(0), f3(0)}, V = span{e1(0), e2(0), e3(0)}.

The differential d� of the projection map � : T ∗M → M definesan identification between ℋ and TxM . On the other hand, the map� : T ∗xM → V defined by �(�) = −� also gives an identification betweenT ∗xM and V .

Let us consider the differential d(dfx) : TxM → TdfxT∗M at x of

the map x 7→ dfx. It defines a three dimensional subspace Λ :=d(dfx)(TxM) of the tangent space TdfxT

∗M . Since Λ is transvesal tothe space V , it defines a linear map S from ℋ to V for which the graphis given by Λ. More precisely, if w = wℎ+wv is a vector in the space Λ,where wℎ and wv are inℋ and V , respectivly. Then S(wℎ) = wv. Underthe identifications of the tangent space TxM with ℋ and the cotangentspace T ∗xM with V , we obtain a linear map HSRf(x) : TxM → T ∗xMcalled subriemannian Hessian. More precisely,

HSRf(x)(d�(w)) = �−1Sw.

Proposition 7.1. The subriemannian Hessian HSRf is symmetric i.e.⟨HSRf(x)v, w

⟩=⟨HSRf(x)w, v

⟩.

Proof. Let w1 and w2 be two vectors in the subspace ℋ. Since thesubspace Λ is a Lagrangian subspace, we have

!(w1 + S(w1), w2 + S(w2)) = 0.

Since both ℋ and V are Lagrangian subspaces, we also have

!(w1, S(w2)) + !(S(w1), w2) = 0.


It follows from skew symmetry of ! and the definition of subrieman-nian Hessian that ⟨

HSRf(x)d�(w2), d�(w1)⟩

= !(w1, S(w2))

= !(w2, S(w1))

=⟨HSRf(x)d�(w1), d�(w2)

⟩.

□

Let �i = d�(fi(0)) and let ℌf(x) be the subriemannian Hessian ma-trix with ij-th entry ℌijf(x) defined by

ℌijf(x) =⟨HSRf(x)�i, �j

⟩= !(Sfi(0), fj(0)).

Recall that v0 denotes the Reeb field and v1, v2 be an orthonormalbasis with respect to the subriemannian metric g.

Proposition 7.2. The subriemannian Hessian matrix ℌf satisfies thefollowing

ℌ11f =(v1f)2v2

2f + (v2f)2v21f − (v1f)(v2f)(v1v2f + v2v1f)

(v1f)2 + (v2f)2

+ a112v2f − a2

12v1f,

ℌ12f = (v1f)v2v0f − (v2f)v1v0f + (v0f)ℌ13f + �1a(df),

ℌ13f =(v1f)(v2f)(v2

1f − v22f)− (v1f)2(v2v1f) + (v2f)2(v1v2f)

(v1f)2 + (v2f)2,

ℌ22f = ((v1f)2 + (v2f)2)v20f − (v0f)(v1f)v1v0f − (v0f)(v2f)v2v0f

+ v0f(ℌ23f + a(df))− �1(df)

ℌ23f = (v0f)ℌ33f − (v1f)v0v1f − (v2f)v0v2f,

ℌ33f =(v1f)2v2

1f + (v1f)(v2f)(v2v1f + v1v2f) + (v2f)2v22f

(v1f)2 + (v2f)2.

Proof. Let A be the matrix with the ij-th entry aij defined by

d(dfx)(�i) = fi(0) +3∑

k=1

aikek(0),

By the definition of the linear map S, we have

S(fi(0)) =3∑j=1

aijej(0).


It follows that

ℌijf(x) = !(S(fi(0)), fj(0)) = −aij.

Let us look at the case a11. We have 2H(df) = (v1f)2 +(v2f)2. Since�(dfx) = x, we also have

�∗�i(d(dfx))(�1) = �i(�1).

Therefore, by Theorem 2.2, we have

ℌijf(x) = −a11

= −!(f1(0), d(dfx)(�1))

=(v1f)2v2

2f + (v2f)2v21f − (v1f)(v2f)(v1v2f + v2v1f)

(v1f)2 + (v2f)2

+ a112v2 − a2

12v1.

Similar calculations give the rest of the entries of A. □

We define the horizontal gradient ∇H by g(∇Hf, v) = df(v) for allvectors v in the distribution Δ. Recall that � is the volume form definedby �(v0, v1, v2) = 1. The sub-Laplacian ΔH is defined by ΔHf =div�∇Hf . Here div� denotes the divergence with respect to the volumeform �. Let C2 be the matrix defined by

C2 =

⎛⎝ 1 0 00 0 00 0 1

⎞⎠ .

Corollary 7.3. The subriemannian Hessian matrix ℌ and the sub-Laplacian satisfies

tr(C2ℌf) = ΔHf,

where tr denote the trace of the matrix.

Proof. A simple calculation using Proposition 7.2 shows that

tr(C2ℌf) = (v21 + v2

2 + a112v2 − a2

12v1)f = ΔHf.

□

Let d be the subriemannian distance function of a subriemannianmanifold (M,Δ, g). Let us fix a point x0 in the manifold M . Letr : M → ℝ be the function r(x) = d(x, x0) and let f(x) = −1

2r2(x).

Finally, we show that the subriemannian Hessian of the function f takesa very simple form.


Proposition 7.4. The subriemannian Hessian matrix ℌf satisfies thefollowing wherever f is twice differentiable

ℌ11f = ΔHf + 1,

ℌ12f = (v1f)v2v0f− (v2f)v1v0f + �1a(df),

ℌ13f = 0,

ℌ22f = −2f v20f + (v0f)

2 − �1(df),

ℌ23f = 0,

ℌ33f = −1.

If we assume that the subriemannian manifold is Sasakian, then theabove simplifies to

ℌ11f = ΔHf + 1,

ℌ12f = (v1f)v2v0f− (v2f)v1v0f,

ℌ13f = 0,

ℌ22f = −2f v20f + (v0f)

2,

ℌ23f = 0,

ℌ33f = −1.

Proof. The first formula follows from differentiating the following equa-tion by v0, v1, and v2

(v1f)2 + (v2f)

2 = −2f

and combining them with Proposition 7.2.The second follows from a = 0. □

8. Sub-Laplacian of Distance Functions in Sasakian SpaceForms

In this section, we give a formula for the sub-Laplacian of the subrie-mannian distance function of a Sasakian space form. Let d be the sub-riemannian distance function of a subriemannian manifold (M,Δ, g).Let us fix a point x0 in the manifold M . Let r : M → ℝ be the functionr(x) = d(x, x0) and let f(x) = −1

2r2(x).

Theorem 8.1. Assume that the subriemannian manifold (M,Δ, g) isa three dimensional Sasakian space form of Tanaka-Webster curvaturek. Then the subriemannian Hessian matrix ℌf satisfies the following


wherever f is twice differentiable.

ℌf =

⎧⎨⎩

−

⎛⎜⎝�(sin �−� cos �)

2−2 cos �−� sin ��2(1−cos �)

2−2 cos �−� sin �0

�2(1−cos �)2−2 cos �−� sin �

�3 sin �2−2 cos �−� sin �

0

0 0 1

⎞⎟⎠ if � > 0,

−

⎛⎜⎝�(� cosh �−sinh �)

2−2 cosh �+� sinh ��2(cosh �−1)

2−2 cosh �+�0 sinh �0

�2(cosh �−1)2−2 cosh �+�0 sinh �

�3 sinh �2−2 cosh �+� sinh �

0

0 0 1

⎞⎟⎠ if � < 0,

−

⎛⎜⎝ 4 6 0

6 12 0

0 0 1

⎞⎟⎠ if � = 0,

where � = (v0f(z))2 − 2f(z)k and � =√∣�∣.

By combining Theorem 8.1 and Proposition 7.3, we obtain the fol-lowing.

Corollary 8.2. Let d be the subriemannian distance function of athree dimensional Sasakian space form of Tanaka-Webster curvaturek. Then the sub-Laplacian ΔHr satisfies the following wherever r istwice differentiable.

ΔHr =

⎧⎨⎩�(sin �−� cos �)

r(2−2 cos �−� sin �)if � > 0,

�(� cosh �−sinh �)r(2−2 cosh �+� sinh �)

if � < 0,4r

if � = 0,

where � = r(z)2((v0r(z))2 + k) and � =√∣�∣.

Proof of Theorem 8.1. Let 't(x) = �(etH(dfx)). Assume that z is apoint where f is twice differentiable. Let Λ be the image of the linearmap d((df)z) : TzM → TdfzT

∗M . Let Et = (e1(t), e2(t), e3(t))T , Ft =(f1(t), f2(t), f3(t))T be a Darboux frame at dfz and let �i = d�(fi(0)).Let At and Bt be the matrices with ij-th entry aij(t) and bij(t), respec-tively, defined by

d(dfy)(�i) =3∑j=1

(aij(t)ej(t) + bij(t)fj(t)) .

We define the matrix St by St = B−1t At. Since �(e1⋅H(dfx)) = x0 for

all x, we have limt→1 S−1t = 0. The same argument as in Theorem 5.1

shows that

St −R + StC1 + CT1 St − StC2St = 0,


where R =

⎛⎝ � 0 00 0 00 0 0

⎞⎠ and � is defined by

� = ℎ0(dfz)2 + 2H(dfz)k = (v0f(z))2 − 2f(z)k = r(z)2((v0r(z))2 + k).

By the proof of Proposition 7.2 and B0 = I, we have ℌf(z) = −A0 =−S0. By the result in [11], we can compute St and it is given by

St =

⎧⎨⎩

⎛⎜⎝�0(sin �t−�t cos �t)2−2 cos �t−�t sin �t

�20 (1−cos �t)

2−2 cos �t−�t sin �t0

�20 (1−cos �t)

2−2 cos �t−�t sin �t

�30 sin �t2−2 cos �t−�t sin �t

0

0 0 11−t

⎞⎟⎠ if � > 0,

⎛⎜⎝�0(�t cosh �t−sinh �t)2−2 cosh �t+�t sinh �t

�20 (cosh �t−1)

2−2 cosh �t+�t sinh �t0

�20 (cosh �t−1)

2−2 cosh �t+�t sinh �t

�30 sinh �t2−2 cosh �t+�t sinh �t

0

0 0 11−t

⎞⎟⎠ if � < 0,

⎛⎜⎝4

1−t6

(1−t)2 06

(1−t)212

(1−t)3 0

0 0 11−t

⎞⎟⎠ if � = 0,

where �t = (1− t)√∣�∣.

By setting t = 0, we obtain the result. □

9. Subriemannian Hessian and Laplacian ComparisonTheorem

In this section, we prove a Hessian and a Laplacian comparison the-orem in our subriemannian setting. Let (M1,Δ1, g1) and (M2,Δ2, g2)be three dimensional contact subriemannian manifolds. Let xi0 be apoint on the manifold M i, let ri be the subriemannian distance fromthe point xi0, and let fi = −1

2r2i . Let vi0 be the Reeb field in M i and

let (R11t )i and (R22

t )i be the curvature invariant on M i introduced insection 2. Finally let

Rit =

⎛⎝ (R11t )i 0 00 (R22

t )i 00 0 0

⎞⎠ .

Theorem 9.1. (Subriemannian Hessian Comparison Theorem I) Letz1 and z2 be points on the three dimensional contact subriemannianmanifolds M1 and M2, respectively, such that fi is twice differentiable

at zi. Assume that R1t

∣∣∣(df1)z1

≤ R2t

∣∣∣(df2)z2

for all t in the interval [0, 1].


Then

ℌf1(z1) ≤ ℌf2(z2).

Remark 9.2. By the result in [8], fi is twice differentiable Lebesguealmost everywhere.

If we restrict to Sasakian manifolds, then we have the following.

Theorem 9.3. (Subriemannian Hessian Comparison Theorem II) Letd be the subriemannian distance function of a Sasakian manifold (M,Δ, g)and let f(x) = −1

2d2(x, x0), where x0 is a point on M . Assume that the

Tanaka-Webster curvature � of M satisfies � ≥ k (resp. � ≤ k). Thenthe following holds wherever f is twice differentiable.

ℌf ≥

⎧⎨⎩

−

⎛⎜⎝�(sin �−� cos �)

2−2 cos �−� sin ��2(1−cos �)

2−2 cos �−� sin �0

�2(1−cos �)2−2 cos �−� sin �

�3 sin �2−2 cos �−� sin �

0

0 0 1

⎞⎟⎠ if � > 0,

−

⎛⎜⎝�(� cosh �−sinh �)

2−2 cosh �+� sinh ��2(cosh �−1)

2−2 cosh �+� sinh �0

�2(cosh �−1)2−2 cosh �+� sinh �

�3 sinh �2−2 cosh �+� sinh �

0

0 0 1

⎞⎟⎠ if � < 0,

−

⎛⎜⎝ 4 6 0

6 12 0

0 0 1

⎞⎟⎠ if � = 0,

(resp. ≤) where � = (v0f)2 − 2kf and � =

√∣�∣.

If we combine Theorem 9.3 and Proposition 7.3, then we have thefollowing sub-Laplacian comparison theorem.

Corollary 9.4. (Sub-Laplacian Comparison Theorem) Under the no-tations and assumptions of Theorem 9.3, the following holds whereverr is twice differentiable.

ΔHr ≤

⎧⎨⎩�(sin �−� cos �)

r(2−2 cos �−� sin �)if � > 0,

�(� cosh �−sinh �)r(2−2 cosh �+� sinh �)

if � < 0,4r

if � = 0,

(resp. ≥)

where r(x) = d(x, x0), � = r(z)2((v0r(z))2 + k), and � =√∣�∣.

Proof of Theorem 9.1. Let 'it(x) = �(etH((dfi)x)) and let

Eit = (ei1(t), ei2(t), ei3(t))T , F i

t = (f i1(t), f i2(t), f i3(t))T


be a Darboux frame at (dfi)zi and let �ij = d�(fj(0)). Let Ait and Bit be

the matrices with jk-th entry aijk(t) and bijk(t), respectively, defind by

d(dfzi)(�j) =3∑

k=1

(aijk(t)eik(t) + bijk(t)f

ik(t)).

We define the matrix Sit by (Bit)−1Ait. As in the proof of Theorem

8.1, we have ℌfi(zi) = −Si0 = −Ai0 and

Sit −Rit + SitC1 + CT

1 Sit − SitC2S

it = 0 lim

t→1(Sit)

−1 = 0,

where Rit here denotes Ri

t

∣∣∣dfzi

.

Therefore, by assumption and the result in [16], we have the followingas claimed

ℌf2 = −S2t ≥ −S1

t = ℌf1.

□

Proof of Theorem 9.3. Let us first assume that � ≥ k. Here we use thesame notation as in proof of Theorem 8.1. The matrix St satisfies theequation

St −Rt + StC1 + CT1 St − StC2St = 0, lim

t→1(St)

−1 = 0,

where

Rt =

⎛⎝ � 0 00 0 00 0 0

⎞⎠ .

By the result in [16], we have St ≤ Skt , where Skt is the solution of

St −Rk + StC1 + CT1 St − StC2St = 0, lim

t→1(St)

−1 = 0,

and

Rk =

⎛⎝ (v0f)2 − 2kf 0 00 0 00 0 0

⎞⎠ .

If we set t = 0, then we get ℌf = −S0 ≥ −Sk0 . Finally the matrix Sk0can be computed using the result in [11] which gives the first statementof the theorem. The reverse inequalities under the assumption � ≤ kare proved in a similar way. □


10. Cheeger-Yau Type Theorem in SubriemannianGeometry

In this section, we give a lower bound on the solution of the sub-riemannian heat equation u = ΔHu in the spirit of its Riemanniananalogue in [9]. More precisely, let � be the function defined by

�(s) =

{√k(sin(s

√k)−s

√k cos(s

√k))

2−2 cos(s√k)−s

√k sin(s

√k)

if k > 04s

if k = 0.

Let ℎ = ℎ(t, s) : [0,∞) × (0,∞) → ℝ be a smooth solution to thefollowing equation

(10.1) ℎ = ℎ′′ + ℎ′�.

where ℎ and ℎ′ denotes the derivative with respect to t and s, respec-tively.

Theorem 10.1. Let (M,Δ, g) be a three dimension Sasakian manifoldwith non-negative Tanaka-Webster curvature. Let ℎ be a solution of(10.1) which satisfies the conditions

ℎ′(0, s) ≤ 0, lims→0

ℎ′(t, s) ≤ 0.

Let x0 be a point on the manifold M and let r(⋅) = d(x0, ⋅), where dis the subriemannian distance function. Let Ω be an open set whichcontains x0 and have smooth boundary ∂Ω. Let u = u(t, x) be a smoothsolution to the subriemannian heat equation u = ΔHu on [0,∞)×M∖Ωwhich satisfies

u(0, ⋅) ≥ ℎ(0, r(⋅)), u(t, y) ≥ ℎ(t, r(y)) y ∈ ∂Ω.

Then we have u ≥ ℎ ∘ r.

Remark 10.2. When k = 0, the function

ℎ(t, s) = (t+ �)−5/2e−s2

4(t+�)

is a solution to the equation (10.1) for every � > 0.

Proof. Let r be the subriemannian distance function from the pointx0 (i.e. r(x) = d(x0, x)). By Corollary 9.4 and the chain rule, thefollowing holds �-a.e.

ΔHr ≤

{�(sin �−� cos �)

r(2−2 cos �−� sin �)if � > 0

4r

if � = 0

where � = r2((v0r)2 + k) and � = r

√(v0r)2 + k.


If we differentiate (10.1) with respect to s. Then we get

(10.2) ℎ′ = ℎ′′′ + ℎ′′�+ ℎ′�′.

By maximum principle, we see that ℎ′(t, s) ≤ 0 for all t and for all ssince ℎ′(0, s) ≤ 0 for all s and ℎ′(t, 0) ≤ 0 for all t by assumptions.

Therefore, the following holds wherever r is twice differentiable.

ΔH(g(t, r)) = ℎ′′(t, r) + ℎ′(t, r)ΔHr

≥ ℎ′′(t, r) + ℎ′(t, r)�(r)

= ℎ(t, r).

(10.3)

Let (t0, z) be a local minimum of the function G(t, x) = u(t, x) −ℎ(t, r(x)) + �t, where � is a positive constant. Let us assume thatt0 > 0 and z in contained in the interior of M∖Ω. By the result in[8], r is locally semiconcave on M∖ {x0}. Since g is nondecreasing ins, G(t, x) is locally semiconcave on M∖ {x0} as well. Therefore, by [7,Theorem 2.3.2], we can find a sequence of points zi on the manifold Mconverging to z and a sequence of numbers �i converging to 0 such that

ΔHG(t0, zi) ≥ −�i.Since u is the solution of the subriemannian heat equation, it follows

from (10.3) that ddtG(t0, zi) ≥ −�i + �. If we let i goes to ∞, then we

have 0 = ddtG(t0, z) ≥ � which is a contradiction.

Since we have the condition u(t, x) ≥ ℎ(t, r(x)) for all points x on theboundary of Ω and u(0, ⋅) ≥ ℎ(0, r(⋅)), it follows that G ≥ 0. Therefore,if we let � goes to 0, then we have u ≥ ℎ ∘ r as claimed. □

References

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[2] F. Baudoin, N. Garofalo: Generalized Bochner formulas and Ricci lowerbounds for sub-Riemannian manifolds of rank two, preprint, arXiv:0904.1623.

[3] F. Baudoin, N. Garofalo: Curvature-dimension inequalities and Ricci lowerbounds for sub-Riemannian manifolds with transverse symmetries, preprint,arXiv:1101.3590.

[4] F. Baudoin, M. Bonnefont, N. Garofalo: A sub-riemannian curvature-dimension inequality, voluem doubling property and the Poincare inequality,preprint, arxiv: 1007.1600.

[5] R. Bhatia: Matrix analysis. Graduate Texts in Mathematics, 169. Springer-Verlag, New York, 1997

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[8] P. Cannarsa, L. Rifford: Semiconcavity results for optimal control problemsadmitting no singular minimizing controls. Ann. Inst. H. Poincare Anal. NonLineaire 25 (2008), no. 4, 773–802

[9] J. Cheeger, S.T. Yau: A lower bound for the heat kernel. Comm. Pure Appl.Math. 34 (1981), no. 4, 465-480

[10] N. Juillet: Geometric inequalities and generalized Ricci bounds in the Heisen-berg group. Int. Math. Res. Not. IMRN 2009, no. 13, 2347-2373

[11] J.J. Levin: On the matrix Riccati equation. Proc. Amer. Math. Soc. 10 1959519–524.

[12] C.B. Li, I. Zelenko: Differential geometry of curves in Lagrange Grassman-nians with given Young diagram. Differential Geom. Appl. 27 (2009), no. 6,723742

[13] C.B. Li, I. Zelenko: Parametrized curves in Lagrange Grassmannians, C.R.Acad. Sci. Paris, Ser. I, Vol. 345, Issue 11, 647–652

[14] C.B. Li, I.Zelenko: Jacobi Equations and Comparison Theorems for Corank1 sub-Riemannian Structures with Symmetries, Journal of Geometry andPhysics 61 (2011) 781–807

[15] R. Montgomery: A tour of subriemannian geometries, their geodesics andapplications. Mathematical Surveys and Monographs, 91. American Mathe-matical Society, Providence, RI, 2002

[16] H.L. Royden: Comparison theorems for the matrix Riccati equation, Comm.Pure Appl. Math. 41 (1988), no. 5, 739-746

[17] S. Tanno: Variational problems on contact Riemannian manifolds. Trans.Amer. Math. Soc. 314 (1989), no. 1, 349-379

E-mail address: [email protected]

International School for Advanced Studies, via Bonomea 265, 34136,Trieste, Italy and Steklov Mathematical Institute, ul. Gubkina 8,Moscow, 119991 Russia

E-mail address: [email protected]

Department of Mathematics, University of California at Berkeley,970 Evans Hall #3840 Berkeley, CA 94720-3840 USA

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BISHOP AND LAPLACIAN COMPARISON THEOREMS ON …BISHOP AND LAPLACIAN COMPARISON THEOREMS ON THREE DIMENSIONAL CONTACT SUBRIEMANNIAN MANIFOLDS WITH SYMMETRY ANDREI AGRACHEV AND PAUL