Small time heat kernel asymptotics at theRiemannian and sub-Riemannian cut locus

Ugo Boscain (CNRS, CMAP, Ecole Polytechnique, Paris)

Davide Barilari, (CNRS, CMAP, Ecole Polytechnique, Paris)

Robert Neel, (Lehigh University)

[1] D. Barilari, U. B. R. Neel, Small time heat kernel asymptotics at thesub-Riemannian cut-locus, to appear on JDG.

November 23, 2012

an old question:

What is the relation between:

(sub)-Riemannian distance←→ small-time heat-kernel asymptotics

the structure of optimal geodesics

In particular, we are interested in what happens at the cut locus

(region where geodesics lose optimality)oC

Riemannian or Sub-Riemannian manifold

dynamite of given power(Dirac-δ)

x y

cut locus

Can we recognize that we are at the cut locus by measuring the heat?

Definition of sub-Riemannian structure

Definition

A sub-Riemannian manifold is a pair (M, {X1, . . . , Xm}) such that{X1, . . . , Xm} satisfies the Hormander condition.

∀ q ∈M, Lieq{X1, . . . , Xm } = TqM

In general dim(M)︸ ︷︷ ︸

n

≥ dim(Span(X1(q), . . . Xm(q)))︸ ︷︷ ︸

k(q)

≤ #of vector fields︸ ︷︷ ︸

m

This definition includes:structure example

Riemannian struct. with M parallelizable n = k = m flat torus

Riemannian struct. with M non-parallelizable n = k < m 2-sphere in R3

Carnot groups n > k = m Heisenberg

equiregular sub-Riemannian struct. n > k ≤ m contact struct.

non-equiregular sub-Riemannian struct. n > k ≤ m Martinet

rank-varying sub-Riemannian struct. n ≥ k(q) ≤ m Grushin

others .....

Define N1(q) := Span(X1(q), . . . Xm(q)), N

i+1 := Ni + [Ni,N]. If

dim(Ni), i = 1, . . . ,m do not depend on the point, it is called equiregular

Horizontal Curves and Carnot-Caratheodory distance

Definition

A Lipschitz continuous curve γ : [0, T ]→M is said to be horizontal if ∃u1(.), . . . , um(.) ∈ L∞([0, T ],R), s.t.

γ(t) =∑m

i=1 ui(t)Xi(γ(t)) for a.e. t ∈ [0, T ].

Definition

The Carnot-Caratheodory distance is

d(q0, q1) = inf{

∫ T

0

√

u21(t) + . . .+ u2

m(t) dt | the corresponding trajectory

joins q0 to q1}

→thanks to the Hormander condition, this distance gives to M a structureof metric space (compatible with its topology) of Hausdorff dimension

Q = n in the Riemannian case

Q = const > n for equiregular sub-Riemannian structures

Q = Q(q) ≥ n in the general case

Laplace operator

Definition

the sub-Riemannian Laplacian is ∆φ := div(gradH(φ)

)

where

gradHφ =∑m

i Xi(φ)Xi

div is the classical divergence computed with respect to a givensmooth volume µ.

Remarks

Second order terms do not depends on µ since

∆φ =m∑

i=1

X2i (φ) + (div(Xi))Xi(φ)

If the structure is equiregular there is a regular intrinsic volume(Popp’s volume) and the corresponding Laplacian is called “intrinsic”.

In the Riemannian case Popp’s volume is the Riemannian volume.

Existence of the heat kernel

Theorem (Hormander, Strichartz)

Consider a sub-Riemannian manifold (M, {X1, . . . , Xk}) which is completeas metric space. Then

the sub-Riemannian Laplacian ∆ w.r.t. a regular volume µ ishypoelliptic (⇐ Hormander condition)

The sub-Riemannian heat equation ∂tφ(t, q) = ∆φ(t, q) admits asmooth kernel pt(x, y) (⇐ completeness)

Computation of Minimizers:

Candidates minimizers are computed via the Pontryagin MaximumPrinciple

normal extremals: projection on q of solutions of

H(q, λ) =1

2

m∑

1

〈λ,Xi(q)〉2

lying on the level set H = 1/2

abnormal extremals satisfy 〈λ,Xi(q)〉 ≡ 0.

→Normal extremals are geodesics:

Definition

a geodesic is a curve γ : [0, T ]→M , parametrized by constant velocity, s.t.for every suff. small interval [t1, t2] ⊂ [0, T ], γ|[t1,t2] is optimal betweenγ(t1) and γ(t2).

→abnormal extremals can be geodesics or not. In this talk I will assumethat there are no abnormal minimizers

In any case one expects that candidate optimal

trajectories loose optimality after some time.

1geodesics

cut locus

Front

conjugate locus

Sphere

3

conjugate locus: where local optimality is lost(the differential of the exponential map is degenerate)

cut locus: where global optimality is lost

sphere(ε): set of points at distance ε from a given point (level sets ofthe value function)

front(ε): end point of geodesics at time ε from a given point

Recall that:

if dim(M) >dim(Span{X1, . . . Xm}) then the cut locus and theconjugate locus are adiacent to the starting point

t2

t

If dim(M) =dim(Span{X1, . . . , Xm}) then they are far from thestarting point

starting point

cut locus

The relation between the distance and the kernel

Can we relate pt(x, y) with d(x, y) ?

What is known in SRG?

Assume that there are no abnormal extremals

On the diagonal.

pt(x, x) =C + o(1)

tQ/2(Ben Arous and Leandre, ’91) (1)

Here Q is the Hausdorff dimension

→pt(x, x) =1

(4πt)n/2 (1 +K(x)

6t+ o(t))

(Riemannian, Minakshisundaram-Pleijel, 1949)

→pt(x, x) =1t2(1 + k(x)

3t+ o(t)) (3D contact, Barilari, 2012)

Off diagonal and off cut locus. Fix x 6= y. If y is not in the cutlocus of x

pt(x, y) =C + o(1)

tn/2e−d2(x,y)/4t (Ben Arous, ’88)

In any point of the space including the cut locus.

limt→0

4t log pt(x, y) = −d2(x, y) (Leandre, ’87) (2)

the gap: what happens on the cut locus?

Specific examples shows that on the cut locus

pt(x, y) =C + o(1)

tre−d2(x,y)/4t with r ≥ n/2

in Riemannian

on S1 we have r = 1/2=n/2

on the cylinder we have that r = 1=n/2

on S2 we have that r = 3/2>1 = n/2(Fischer, Jungster, and Williams, 1984)

→why this difference?

in sub-Riemannian

on the Heisenberg group on the z axis we have r = 2>3/2 = n/2(by Gaveau 1977)

But the problem was open for 20 years.

reasons ???

absence of results in the Riemannian case(however there was a pioneering ideas of Molcanov in ’75 that wasoverlooked)

no information on the cut locus in sub-Riemannian geometry besidesthose on the Heisenberg group and symmetric nilpotent (n, n+ 1)groups

Now we have a better understanding of the cut locus in sub-Riemanniangeometry (at least in STEP 2)

Complete results on:

local structure in 3D contact (Agrachev, Gauthier and Kupka, ’96)

SU(2), SO(3), Sl(2) with the metric induced by the Killing form(Francesco Rossi and U.B., 2009)

SE(2) by Yuri Sachkov (2010-2011)

non-symmetric nilpotent (4,5) case (Barilari, U.B. 2013)

Partial results on

nilpotent (4,10) by Brockett (????)

nilpotent (3,6) by Myasnichenko (2002)

nilpotent (2,3,4) (2,3,5) Yuri Sachkov (2004)

Heat-kernel asymptotic at the cut locus

Announcement

Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds

IHP, Paris, Sep-Dec 2014

Organizing Committee: A. Agrachev, U.B, Y. Chitour, F. Jean, M.Sigalotti, L. Rifford

Scientific Commitee: A. Agrachev, L. Ambrosio, U. Boscain, Y. Chitour,R. Bryant, E. Falbel, A. Figalli, B. Franchi, J.P. Gauthier, N. Garofalo, F.Jean, I. Kupka, A. Malchiodi, R. Montgomery, P. Pansu, J. Petitot, L.Rifford, A. Sarychev, F. Serra Cassano, M. Sigalotti, E. Trelat, I. Zelenko.

4 courses at M2 level

4 workshops

several thematic days

many seminars

→there will be the possibility of financing students

→we have money for several invitations. We are looking for more ...

→www.cmap.polytechnique.fr/subriemannian

The Molcanov technique

(how to get information on the heat kernel asymptotic at the cut locus)

→Assume that there are no abnormal minimizers.

By the semi-group property (or Chapman-Kolmogorov equation, forprobabilists), we have

pt(x, y) =

∫

M

pt/2(x, z)pt/2(z, y)µ(dz)

Let Γ the set of midpoints of the geodesics going from x to y.(cut locus)

Γ

Γ

y

cut locus

y

xx

pt(x, y) =

∫

N(Γ)

pt/2(x, z)pt/2(z, y)µ(dz) +

∫

M\N(Γ)

pt/2(x, z)pt/2(z, y)µ(dz)

N(Γ)N(Γ)

First term IN(Γ) =∫N(Γ) pt/2(x, z)pt/2(z, y)µ(dz)

On N(γ) there are no cut points neither from x neither from y ⇒ we canuse the Ben Arous expansion

pt(x, z) =1

tn/2e−d2(x,z)/4t(C1(x, z) +O(t)), pt(z, y) =

1

tn/2e−d2(z,y)/4t(C2(z, y) +O(t))

Then

IN(Γ) =

∫

N(Γ)

1

tne−

d2(x,z)+d2(z,y)4t (C(x, y, z) +O(t))µ(dz)

=

∫

N(Γ)

1

tne

−hx,y(z)

t (C(x, y, z) +O(t))µ(dz)

Where hx,y(z) =d2(x,z)+d2(z,y)

4is called the Hinged energy function.

Now

For t small only the behaviour of hx,y(z) around its minimum isimportant (Laplace integral).

For the same reason∫

M\N(Γ)is small

the hinged energy function and its minimum

hx,y(z) =d2(x, z) + d2(z, y)

4

yΓ

z d(z, y)d(x, z)

x

Lemma

hx,y(z) obtains its minimum exactly on Γ and it is smooth in aneighborhood of Γ.

The analysis of the asymptotic IN(Γ) permits to obtain

Theorem (Barilari, U.B., Neel)

Assume that there is only one optimal geodesic from x to y. If there exists acoordinate system around z0 such that

hx,y(z) =1

4d2(x, y) + z2m1

1 + . . .+ z2mnn + o(|z1|

2m1 + . . .+ |zn|2mn ) (3)

for some integers 1 ≤ m1 ≤ m2 ≤ · · · ≤ mn then

pt(x, y) =C + o(1)

tn−

∑i

12mi

exp

(

−d2(x, y)

4t

)

. (4)

If the minimum is not degenerate then by Morse Lemmahx,y(z) =

14d2(x, y) + z21 + . . .+ z2n.

In this case one gets tn−n 12 = tn/2

If the number of minimal geodesics connecting x to y is not one butfinite one gets sever contributions of the kind above

If there exists a one (or more) parameter family of optimal geodesicsjoining x to y and coordinates such that hx,y does not depend oncertain variables. Then some mi = +∞.

What is the relation among the expansion of hx,y(z) and the properties ofoptimal geodesics joining x to y?

Recall that geodesics are projections of solution to the Hamiltonian systemdefined by H(q, λ) =

∑〈λ,Xi(q)〉

2 corresponding to the level set 1/2.

Define the exponential map Ex map as follows:

(λ0, t) ∈ T ∗xM∩{H = 1/2}×R+ → {projection of the solution starting from (x, λ0)}

Properties:

For every λ0, γ(t) = Ex(λ0, t) is a geodesic parameterized by arclength.

Ex(λ0, t) depends only on the product λ0 t i.e. We can consider it as amap from T ∗

xM to M .

The first conjugate time of is tcon(γ) = min{t > 0, (λ0, t) is a criticalpoint of Ex}.

Conjugacy of Ex and Degeneracy of Hessz0hx,y(z)

Theorem (Barilari, U.B., Neel)

x and y are conjugate along γ if and only if the Hessian of hx,y at z0is degenerate.

In particular γ is conjugate in the direction λ′(0) (i.e.ddsEx(2λ(s))|s=0 = 0) if and only if the Hessian of hx,y at z0 is

degenerate in the corresponding direction z′(0) (i.e. z′(0) ifd2

ds2hx,y(z(s))|s=0 = 0).

The dimension of the space of perturbations for which γ is conjugate isequal to the dimension of the kernel of the Hessian of hx,y at z0.

z′(0)

y = Ex(2λ)

Ex(2λ(s))

x

T ∗xM

λ(s)

λ′(0) 6= 0

λ=λ(0)

Γ

γ

z0 = Ex(λ)

z(s) = Ex(λ(s))

The main result

Theorem (Barilari, U.B., Neel)

(less degenerate case): when x and y are not conjugate:

hx,y(z) =1

4d2(x, y) + z21 + . . .+ z2n + o(|z1|

2 + . . .+ |zn|2), and

pt(x, y) =C +O(t)

tn/2e−d2(x,y)/4t,

(most degenerate case): when the only non degenerate direction is t:

hx,y(z) =1

4d2(x, y) + z21 + o(|z1|

2) and

pt(x, y) =C +O(t)

tn−(1/2)e−d2(x,y)/4t

when the degeneration is only in one direction and it is “minimal”:

hx,y(z) =1

4d2(x, y) + z21 + . . .+ z2n−1 + z4n + o(|z1|

2 + . . .+ |zn|4), and

pt(x, y) =C +O(t)

t(n/2)+(1/4)e−d2(x,y)/4t, (5)

the case of a Riemannian surface (with G. Charlot)

For a generic conjugate point on a surface we get

Pt(x, y) =C +O(t)

t2−(1/2+1/4)e−d2(x,y)/4t =

C +O(t)

t5/4e−d2(x,y)/4t

This was not known even for the Ellipsoid (see Barilari-Jendrej, 2012)

C+o(1)t1

e−d2(x,y)/4t

conjugate

C+o(1)

t5/4e−d2(x,y)/4t

cut

the local 3D contact case (with G. Charlot)

For a generic conjugate point in 3D contact we get

Pt(x, y) =C +O(t)

t3−(1/2+1/2+1/4)e−d2(x,y)/4t =

C +O(t)

t7/4e−d2(x,y)/4t

conjugate

cut

generic 3D contact

Heisenberg

C+o(1)

t3/2e−d2(x,y)/4t

C+o(1)

t7/4e−d2(x,y)/4t

Remarks

In general h is not “diagonalizable” and there are mixed terms.

case with abnormals ???

Grushin-Baouendi: X1 = (1, 0), X2 = (0, x)

Cut

-6 -4 -2 2 4

-10

-5

5

10 Cut

Cut

-1.0 -0.5 0.5 1.0

-0.3

-0.2

-0.1

0.1

0.2

0.3

µ = dx dy

∆ = ∂2x + x2∂2

y

pt(q, q′) pt(q, q

′)q Riemannian point q degenerate point

diagonal(Leandre Ben Arous) ∼ C

t∼ C

t3/2

off diagonal off cut

(Ben Arous) ∼ Cte−d2(q,q′)/(4t) ∼ C

te−d2(q,q′)/(4t)

off diagonal

cut (non-conjugate) ∼ Cte−d2(q,q′)/(4t) ∼ C

te−d2(q,q′)/(4t)

off diagonal —

cut conjugate ∼ C

t5/4e−d2(q,q′)/(4t) (no cut conjugate)

Recall that if µ is the Riemannian volume 1|x|

dx dy then theLaplace-Beltrami operator for the Grushin metric is

∆ = ∂2x + x2∂2

y −1

x∂x

which is essentially self-adjoint on the half plane. Hence no heat is passingthrough the Grushin set. [C. Laurent, U.B. Annales Institut Fourier, toappear]

Thanks