HEISENBERG SUBLAPLACIAN AND THE GRUSHIN OPERATOR
DER-CHEN CHANG AND YUTIAN LI
Abstract. The subLaplacian on the Heisenberg group and the Grushin operator are typical examples of subelliptic operators. Their heat kernels are both given in the form of Laplace type integrals. By using Laplace’s method, the method of stationary phase and the method of steepest descent, we derive the small-time asymptotic expansions for these heat kernels, which are related to the geodesic structure of the induced geometries. Key words: heat kernel, Heisenberg group, Grushin operator, small-time asymptotics, saddle point method, geodesic, conjugate point
MS Classification(2010): Primary: 35K08; Secondary: 53B21
1. Introduction
1.1. Heat kernel asymptotics for the Laplace-Beltrami operator. It is well-known that much geometric information about a Riemannian manifold can be decoded from the small-time asymptotic expansions of the heat kernel (i.e., fundamental solution of the heat operator) of the associated Laplace-Beltrami operator. Let (M, g) be a Riemannian manifold of n-dimension, ρ(q0, q) be the distance between two points q0, q ∈ M, and ∆
be the Laplace-Beltrami operator, and p(t, x, y) be the heat kernel of ∆. The celebrated result of Varadhan [31, 32] reads
(1.1) lim t→0+t log p(t, q0, q) = − 1 2ρ 2(q 0, q).
More precise small-time asymptotic approximations for p(t, q0, q) were investigated by
many authors. As t→ 0+, p(t, q
0, q) has the following expansion
(1.2) p(t, q0, q)∼ 1 (2πt)n/2e −1 2tρ 2(q 0,q)a 0(q0, q) + a1(q0, q)t1/2+ a2(q0, q)t +· · · ,
for q0 and q are near points such that they are joined by a finite number of shortest
geodesic along which they are not conjugate. The half-integer power terms vanish for manifold without boundary. If q0 and q are conjugate to each other along the shortest
geodesic, the asymptotic behavior of p(t, q0, q) will be different, namely, the leading power
of t in the expansion changes from t−n/2 to t−(n+k)/2 for some positive number k. The number k appearing in the power is different for different situations. The details can be found in Molchanov’s survey paper [26].
In the case when M is compact, consider the heat kernel trace,
(1.3) trace (p) = Z M p(t, q, q)dV (q) = ∞ X j=1 e−tλj,
where 0 < λ1 ≤ λ2 ≤ · · · are the eigenvalues of the Laplace-Beltrami operator with
vanishing Dirichlet condition if M has a nonempty boundary. M. Kac’s famous question “can one hear the shape of a drum” [20] concerns the extraction of geometric information
of M from the asymptotic expansion of the heat kernel trace. McKean-Singer [24] showed that (1.4) ∞ X j=1 e−tλj ∼ C0 tn/2 + C1 t(n−1)/2 + C2 t(n−2)/2 +· · · , t→ 0 +,
where C0, C1 and C2 are all global geometric quantities given by
(1.5) C0 = vol(M ) (2π)n/2, C1 = A(∂M ) 4(2π)(n−1)/2, C2 = R M R(x)dx 6(2π)(n−2)/2,
with vol(M ), A(∂M ) and R(x) being the volume of M , the volume of boundary of M , and the scalar curvature, respectively. By the Hardy-Littlewood tauberian theorem, the leading term in the expansion (1.4) yields the classical Weyl’s asymptotic formula [35] for large eigenvalues
(1.6) λn∼ n vol(M ) 2/n , n → ∞.
1.2. Heat kernel asymptotics for subelliptic operators. On an n-dimensional Rie-mannian manifold M , one needs n independent smooth vector fields{X1, X2,· · · , Xn} to
introduce a metric g given by the n×n positive definite matrix (g(Xi, Xj))n×n = (gij)n×n,
and the Laplace-Beltrami operator is given by
(1.7) ∆ = 1 2(det g) 1/2 n X i,j=1 ∂ ∂xi gijg−1/2 ∂ ∂xj ,
which is an elliptic operator. Here, (gij) is the inverse matrix of (gij). When one or several
vector fields are missing, say, given{Xj}kj=1 with k < n, the possible generalizations of the
elliptic operators, Riemannian geometries and their relations are of particular interest. H¨ormander [18] studied a class of operators called “sum of squares of vector fields” of the form (1.8) L = 1 2 k X j=1 Xj2, with Xj = n X i=1 aij(x) ∂ ∂xi,
where ai(x)’s are smooth functions on M . H¨ormander showed that L is subelliptic (and
hence hypoelliptic) if these vector fields satisfy the bracket-generating condition. The hypoellipticity means that Lu = f has a smooth solution provided that f is smooth.
On the other hand, given a bracket generating vector fields {Xj}kj=1 with k < n, we
can introduce a subRiemannian metric on M . The definition of the bracket generating condition and a brief description of the subRiemannian space (M,D, g) are provided in Appendix A.
It seems that subelliptic operators and subRiemannian geometries are natural general-izations of elliptic operators and Riemannian geometries. Now we return to our question, could one obtain geometric information of (M,D, g) from the small-time asymptotics of the heat kernel of L?
Leandre [21, 22] proved a result for the subelliptic heat kernels
(1.9) lim t→0+t log p(t, q0, q) = − 1 2d 2(q 0, q).
Here, d(q0, q) denotes the subRiemannian distance between q0 and q. This result could
be regarded as a generalization of Varadhan’s (1.1) to subelliptic operators. A refined asymptotic formula was then given by Ben Arous [4], who showed that
(1.10) p(t, q0, q)∼
1 tn/2e
−d2(q0,q)2t a
0(q0, q) + O(t1/2) , t → 0+.
for q0 6= q and q is not on the cut-locus of q0. This is an analogue to the elliptic case
given in (1.2). The above results suggest that subelliptic heat kernels have the similar small-time behavior as elliptic ones. However, the subelliptic operators also show some new phenomena. On the diagonal, i.e. when q0 = q, Ben Arous and Leandre [5,6] proved
(1.11) p(t, q, q)∼ C
tQ/2 + O(t
(1−Q)/2), t→ 0+.
The asymptotic behavior of p(t, q0, q) is not known for q 6= q0 and q is a cut point of q0.
The recent breakthrough is given by Barilari, Boscain and Neel [2], who showed that (1.12) p(t, q0, q)∼ 1 (2πt)(n+k)/2e −1 2td 2(q 0,q){a 0(q0, q) + o(1)} , t → 0+,
if q 6= q0 and q is a cut point as well as a conjugate point of q0 along some shortest
geodesic. Here, k is a positive number that reflects ‘how conjugate’ the two points are, in particular, when there is a k-parameter family of shortest geodesics. This result can be regarded as an extension of Molchanov’s result [26] on elliptic operators to subelliptic operators. The investigations on the heat kernel asymptotics for some subelliptic heat kernels are also carried out in detail by Brockett and Mansouri [7] and S´eguin and A. Mansouri [30].
Note that only the first term is derived in the above results on small-time asymptotic behavior of the subelliptic heat kernel. The results for the elliptic operator in (1.2) and (1.4)-(1.5) suggest the higher-order terms in the heat kernel asymptotic expansion will give interesting geometric quantities of the associated subRiemannian geometry. In the present paper, we shall use two examples as an illustration of how to derive the higher order terms in the heat kernel asymptotic expansions.
1.3. Heat kernels for the Heisenberg subLaplacian and the Grushin opera-tor. Two typical and simple examples of subelliptic operators are the subLaplacian on Heisenberg group and the Grushin operator.
The m-th Heisenberg group is a nilpotent Lie group of step two on the manifold (1.13) Hm ∼= Cm× R = {(z, y) = (z1, z2,· · · , zm, y) : z ∈ Cm, y ∈ R},
with the group law
(1.14) (z, y)◦ (w, s) =z + w, y + s + 2 Im m X j=1 ajzjwj ,
where aj’s are positive parameters. For the sake of simplicity, we restrict ourselves on
H1 ={(x1 + ix2, y)}, and assume a1 = 1/2 without loss of generality. The vector fields
(1.15) X1 = ∂ ∂x1 + x2 ∂ ∂y, X2 = ∂ ∂x2 − x 1 ∂ ∂y,
are left-invariant under the group law and bracket-generating, that is [X1, X2] = 2∂y∂
recovers the missing direction. Therefore, the subLaplacian
(1.16) ∆H = 1 2(X 2 1 + X 2 2),
is subelliptic by H¨ormander’s theorem. The heat kernel of ∆H is well studied in literatures
(see [15, 19, 3]). Thanks to the group structure, we can fix q0 to be the origin and let
the other point q(x1, x2, y) vary, and the heat kernel pt(x1, x2, y) is given as a Laplace
integral (1.17) pt(x1, x2, y) = 1 (2πt)2 Z ∞ −∞ e−f (τ )/tV (τ ) dτ,
where the phase function (also known as the modified complex action function) is (1.18) f (τ ) =−iτy + 1
2kxk
2
τ coth τ, with kxk2 = x21+ x22, and the amplitude function (also known as the volume element) is
(1.19) V (τ ) = τ
sinh τ.
It is shown in [3] that the critical points of f (τ ) and the geodesics joining q(x1, x2, y) and
the origin are one-one correspondence, and each of the critical value of f (τ ) is equal to half of square of the length of corresponding geodesic.
The second example of subelliptic operators is introduced by Grushin [17]. Consider the following vector fields on Rm+1 ={(x
1, x2,· · · , xm, y)} (1.20) Xj = ∂ ∂xj , Yj = xj ∂ ∂y, 1≤ j ≤ m.
These vector fields give all (m + 1) directions on Rm+1 except on y-axis, where their bracket [Xj, Yj] = ∂y∂ gives the missing direction. The Grushin operator
(1.21) ∆G = 1 2 m X j=1 (Xj2+ Yj2) = 1 2 m X j=1 ∂2 ∂x2j + 1 2(x 2 1+ x 2 2+· · · + x 2 m) ∂2 ∂y2
is therefore subelliptic by H¨ormander’s theorem. When m = 1, ∆Gis the classical Grushin
operator and its heat kernel is constructed in [11], the general case of m ≥ 1 is studied in [10]. The heat kernel of ∆G has the following form
(1.22) pt(q0, q) = 1 (2πt)(m+2)/2 Z ∞ −∞ e−g(τ )/tW (τ ) dτ,
where q0(x0, y0) and q(x, y) are two points in Rm+1, the phase function (the modified
complex action function) is (1.23) g(τ ) =−i(y − y0) + τ 2 sinh τ (kxk 2+ kx0k2) cosh τ− 2hx, x0i and the amplitude function (the volume element) is
(1.24) W (τ ) =
τ sinh τ
m2 .
Note that the heat kernels (1.17) and (1.22) are both Laplace type integrals. To derive their small-time asymptotic expansions, we can use the techniques of integrals such as Laplace’s method, the method of stationary phase and the method of steepest descent.
The aim of the present paper is to use these techniques to establish the asymptotic expansions for the heat kernels (1.17) and (1.22), and to compute the first few coefficients. This will shed some light on the connection between the heat kernel asymptotics of the subelliptic heat kernel and the subRiemannian geometry.
The remaining part of this paper is organized as follows.
2. Asymptotics for the heat kernel of the Heisenberg subLaplacian For the heat kernel (1.17) of the Heisenberg group, we have the following result. Theorem 1. The heat kernel pt(x1, x2, y) of the Heisenberg group given in (1.17) has the
following asymptotic expansions as t→ 0+:
(I) when (x, y) = (0, 0),
(2.25) pt(0, 0, 0) =
1 8t2;
(II) when (x, y) = (0, y) with y 6= 0,
(2.26) pt(0, 0, y)∼ 1 2t2 ∞ X k=1 e−kπyt (−1)k+1k; (III) when x6= 0, (2.27) pt(x1, x2, y)∼ 1 2π2t3/2e −d2(x1,x2,y)2t ∞ X n=0 Γ n + 1 2 Dntn,
where d(x1, x2, y) is the subRiemannian distance between the origin and the point (x1, x2, y),
and the coefficients Dn = Γ(n + 12)C2n with Cn given in (2.35) for y = 0 and in
(2.51)− (2.52) for y6= 0.
2.1. Case I: diagonal, i.e., x1 = x2 = y = 0. In this case, f (τ ) = 0 and the integral in
(1.17) reduces to (2.28) pt(0, 0, 0) = 1 (2πt)2 Z ∞ −∞ τ sinh τdτ = 1 8t2.
2.2. Case II: x = (x1, x2) = (0, 0) with y 6= 0. Without loss of generality, we assume
y > 0. Now the heat kernel is simply given as
(2.29) pt(0, 0, y) = 1 (2πt)2 Z ∞ −∞ eiτ y/t τ sinh τdτ,
which can be evaluated via the residue calculation. Note that each ikπ, k = 1, 2, . . . , is a simple pole of V (τ ) with the residue being e−kπyt (−1)kikπ. Then, we have
(2.30) pt(0, 0, y) = 1 (2πt)2 ∞ X k=1 e−kπyt (−1)k+12kπ2.
Note that kπy = `2k/2 with `k denoting the length of the k-th geodesic joining (0, 0, y)
and the origin so that `1 gives the subRiemannian distance between these two points.
Therefore, the leading term asymptotic approximation reads pt(0, 0, y)∼
1 (2πt)2e
−d2(0,0,y)2t
+ exponentially small terms, t→ 0+.
2.3. Case III-1: y = 0. Now, the heat kernel (1.17) takes the form (2.31) pt(x1, x2, y) = 1 (2πt)2 Z ∞ −∞ e−τ coth(τ )kxk22t τ sinh τdτ.
Since the exponent f (τ ) is real, the asymptotic expansion of the last integral as t→ 0+
can be derived by Laplace’s method. For a description of this method; see [34]. We recall some properties of the phase function (the modified complex function)
f (τ ) = 1 2kxk
2τ coth(τ ).
The function f (τ ) is real positive and has only one minimum point at τ = 0. That is f0(τ ) = 0 ⇔ τ = 0. This critical point corresponds to the unique geodesic connecting the point (x1, x2, 0) and the origin (0, 0, 0), which is the line segment joining these two
points. Moreover, f (0) = 1 2kxk 2 = 1 2d 2(x 1, x2, 0),
where d(x1, x2, 0) denotes the subRiemannian distance between the point (x1, x2, 0) and
the origin. Now, f and V have the following Taylor’s expansions at the point τ = 0:
(2.32) f (τ ) = 1 2kxk 2 τ coth τ = ∞ X k=0 αkτk (2.33) V (τ ) = τ sinh τ = ∞ X k=0 βkτk, where α0 = kxk 2 2 , α2 = kxk2 6 , α4 =− kxk2 90 , α6 = kxk2 945 , · · · , α2j+1 = 0, β0 = 1, β2 =− 1 6, β4 = 7 360, β6 =− 31 15120, · · · , β2j+1 = 0. By Laplace’s method, we have the small t asymptotic expansion
(2.34) pt ∼ 1 (2πt)2e −f (0)t ∞ X n=0 2Γ n + 1 2 Cnt n+1 2 ∼ 1 2π2t3/2e −kxk2 2t ∞ X n=0 Γ n + 1 2 Cntn/2
where the coefficients Cn can be expressed in terms of αk and βk with 0 ≤ k ≤ n. The
first few coefficients are given by (2.35) C0 = β0 α0 = 2 kxk2, C1 = 0, C2 = 1 α3 0 β2− 3α2β0 α0 =− 28 3kxk6, C3 = 0;
see [34]. Other coefficients can be obtained iteratively by the method in [33] or [23, eq. (28)].
2.4. Case III-2: x 6= 0 and y 6= 0. As t → 0+, the heat kernel in (1.17) is a Laplace
type integral with 1/t playing the role of a large parameter. Note that the action function f (τ ) is analytic in τ . Hence, we may apply Debye’s method of steepest decent to derive the asymptotic expansion; see [34, Sec. II.4]. The basic idea is to deform the integration path to the steepest decent curve Γ, so that the following conditions hold:
(a) Γ passes the critical points of f (z), i.e. points such that f0(z) = 0;
(b) the imaginary part of f (τ ) is a constant along Γ, (here Im f (z) = 0 on Γ).
The term steepest descent stems from condition (b) above, since −Re f(τ) is steepest descent along the curve Γ.
We first recall the results on f (τ ) in [3]. For (x1, x2) 6= (0, 0), f(τ) has finitely many
of critical points, all are purely imaginary. Denote these saddle points as τj = iθj,
j = 1, 2, . . . , p, with 0 < θ1 < θ2 <· · · < θp, see Fig. 1 (b). It is known that
(2.36) f (τj) =
1 2`
2
j, j = 0, 1, . . . , p,
where `j is the length of the j-th geodesic joining (x1, x2, y) and the origin. In particular,
the first critical point gives the smallest critical value, f (τ1) =
1 2d
2
(x1, x2, y)
with d(x1, x2, y) = `1 denoting the subRiemannian (Carnot-Carath´eodory) distance
be-tween (x1, x2, y) and the origin. Since other critical values are larger than the first one,
the contributions of these critical points are exponentially small compared with the first critical point. Write τ = a + ib with a, b both real. The imaginary part of f (τ ) defined in (1.18) is
(2.37) Im f (τ ) = −ay +1 2kxk
2−a sin b cos b + b sinh a cosh a
sinh2a + sin2b .
Note that Im f (τ1) = Im f (iθ1) = 0. The steepest descent path is plotted in Figure 1,
where we choose kxk2 = 2 and y = 5 for illustration: there are three critical points
τj = iθj, j = 1, 2, 3 with 0 < θ1 < π < θ2 < θ3 < 2π. They are the intersecting points of
the steepest path with the imaginary axis.
The Taylor expansions of f and V and the first saddle point τ = τ1 are
(2.38) f (τ ) = ∞ X k=0 ak(τ − τ1)k, V (τ ) = ∞ X k=0 bk(τ − τ1)k+α−1,
with α = 1. By the method of steepest descent, we have
(2.39) pt(x1, x2, y)∼ e−d22t (2πt)2 ∞ X n=0 CnΓ n + 1 2 tn+12 ∼ e −d2 2t (2π)2t32 ∞ X n=0 CnΓ n + 1 2 tn2,
where the coefficients Cn can be derived iteratively; see [34, §II.4]. We follow the method
described in [23] to obtain these coefficients Cn. A simple calculation yields
(2.40) f0(τ ) =−iy + i1 2µ(−iτ)kxk 2 and f00 (τ ) = 1 2µ 0 (−iτ)kxk2,
Reτ Imτ π 2π 3π O τ1 τ2 τ3 Imf2(τ ) = 0 Imf (τ ) = 0 θ µ(θ) O π 2π 3π 2y kxk2 θ1 θ2 θ3
Figure 1. (a) Complex τ plane: the blue curve is the steepest descent path for f (τ ); the red line is the steepest descent path for f2(τ ). (b)
The function µ(θ): the solutions of µ(θ) = 2y/kxk2 give the saddle points τj = iθj. Parameters: kxk2 = 2, y = 5.
where
(2.41) µ(ϕ) = ϕ
sin2ϕ− cot ϕ;
see [3, eq. (1.26)]. By Lemma 1.33 of [3], we have µ0(ϕ) > 0 for 0 < ϕ < π, and hence f(2)(τ1) > 0, which implies that τ = τ1 is a saddle point of order one. Moreover, we also
have f(3)(τ 1) > 0. Thus, f (τ ) = f (τ1) + f(2)(τ1) (τ − τ1)2 2 + f (3)(τ 1) (τ − τ1)3 6 +· · · . Follow the notations of [23],
(2.42) m = min{k > 1 ; f(k)(τ1)6= 0}, p = min{k > m ; f(k)(τ1)6= 0}.
In our case m = 2 and p = 3. Now, we split the phase function into two terms (2.43) f (τ ) = f2(τ ) + τ3f3(τ ), with (2.44) f2(τ ) =: f (τ1) + f(2)(τ 1) 2! (τ − τ1) 2, (2.45) f3(τ ) := f (τ )− f2(τ ) τ3 = f(3)(τ 1) 3! + f(4)(τ 1) 4! (τ − τ1) +· · · .
The idea used in [23] is to deform the contour into a steepest descent path of f2(τ ) instead
and Im f2(τ1) = Im f (τ1) = 0, so the steepest descent path for f2(τ ) is the horizontal line,
which is plotted in Figure1with a red line. This contour is divided into two semi-infinite lines
(2.46) Γk =τ ∈ C; τ = τ1 + reikπ, r≥ 0 , k = 0, 1.
Then the integral in (1.17) becomes an Laplace integral over r, and any method for computing the coefficients of Laplace’s methods can be applied. Follow the approach in [23], the coefficients Cn in the expansion (2.39) are given by
(2.47) Cn= cn(0) + cn(1),
with cn(1) = (−1)ncn(0) and the Laplace’s coefficients cn(0) being calculated recursively
by Wojdylo’s method [33, p. 71] (2.48) cn(0) = 1 mΓ(n+αm ) n X k=0 bn−k k X s=0 (−1)s a(n+α)/m+sm Bk,s s! Γ n + α m + s .
Here, bk is given in (2.38) and the partial ordinary Bell polynomials Bn,k are defined by
(2.49) B0,0 = 1, Bn,0 = 0, Bn,k = n−p+m X j=k−1 an+m−jBj,k−1, n ≥ k ≥ 1. Recall that (2.50) an= 1 n!f (n) (τ1) =− (−i)n 2n! µ (n−1) (−iτ1)kxk2, n≥ 2.
The first few terms of Bn,k are given by
B1,1 = 1 4kxk 2 µ0, B2,1 = −i 12kxk 2 µ00, B2,2 = 1 16kxk 4 (µ0)2, B3,1 = −1 48kxk 2µ000 , B3,2 = −i 24µ 0 µ00, B3,3 = 1 64kxk 6(µ0 )3.
with µ0 := µ0(−iτ1), µ00:= µ00(−iτ1) and µ000 := µ000(−iτ1). A slight calculation yields
(2.51) C0 = b0 a1/22 = τ1 sinh τ1 2 kxk s 1 µ0(−iτ 1) = θ1 kxk r 2 sin θ1 sin θ1 − θ1cos θ1 , (2.52) C2 = b2− 32b1− 32aa3 2b0+ 15 8 b0 a3/22 (2.53) C2j+1 = 0, j = 0, 1, 2, 3,· · · , where a2 = sin θ1− θ1cos θ1 2 sin3θ1 kxk 2, a 3 =
θ1+ 2θ1cos2θ1+ 3 sin θ1cos θ1
6 sin4θ1 kxk 2, b0 = θ1 sin θ1 , b1 = sin θ1− θ1cos θ1 sin2θ1 , b2 =
θ1(1 + cos2θ1)− 2 sin θ1cos θ1
sin3θ1
2.5. Some observations. We have the following form of asymptotics for the heat kernel (2.54) pt(x1, x2, y)∼ C tQ/2e −d2 2t,
where C and Q are constants and d is the Carnot-Carath´eodory distance between (x1, x2, y)
and the origin. We note that the power of t, α, varies. Namely, (2.55) 2α = 4 = Q > n, when x = 0, y = 0, diagonal;
4 = n + 1, when x = 0, y6= 0, off-diagonal and cut-conjugate; 3 = n, when x6= 0, off-diagonal and not cut-conjugate. Here, n = 3 is the topological dimension and Q is the Hausdorff dimension; cf. eq. (A.81). This agrees with the previous result on the asymptotics for the heat kernels on the diagonal, i.e. when the (x1, x2, y) = (0, 0, 0); see [2, 4, 5, 6].
3. Asymptotics for the heat kernel of the Grushin operator
The geodesic structure of the Grushin plane is similar to the Heisenberg case. However, we do have exceptional geodesics in Grushin case, which do not correspond to the critical points of g(τ ) but the singularities of W (τ ), see [10,11,12]. All the geodesics correspond to τk = iηk with 0 < η1 ≤ η2 ≤ η3 ≤ · · · being a (finite or infinite) sequence of positive
numbers. An exceptional geodesic corresponds to ηk = jπ for some positive integer j
appears when (i) x = x0 6= 0 or (ii) x = −x0 6= 0. Moreover, the first geodesic is always
generic in case (i), i.e. 0 < η1 < π≤ η2. However, the first geodesic might be exceptional
in case (ii), that is η1 = π = η2 or η1 = π < η2.
In this section, we shall derive the following result for the Grushin heat kernel with m = 1.
Theorem 2. The heat kernel pt(x0, y0; x, y) of the Grushin operator given in (1.22) with
m = 1 has the following asymptotic expansions as t→ 0+: (I) when x = x0 = 0 and y = y0,
(3.56) pt(x, y; x, y) = 1 (2πt)3/2 Z ∞ −∞ τ sinh τ 12 dτ ; (II) when x = x0 6= 0 and y = y0,
(3.57) pt(x, y; x, y)∼ 2 (2π)3/2t ∞ X s=0 Γs + 1 2 c2sts, where cs is given in (3.61);
(III) when x =−x0 6= 0 and |y − y0| = π2x2,
(3.58) pt(x0, y0; x, y)∼ e−d2(q0,q)2t (2π)3/2t5/4 ∞ X s=0 Γs + 1 4 Dsts;
(IV) other cases,
(3.59) pt(x0, y0; x, y)∼ e−d2(q0,q)2t (2π)3/2t ∞ X s=0 Γ s + 1 2 Dsts,
where d(q0, q) is the subRiemannian distance between the points q0(x0, y0) and q(x, y),
Ds = C2s, and Cs can be computed by formulas in (2.47)-(2.49) with m = 2, p = 3,
α = 1 and as, bs given in (3.73).
When x = x0 = 0, y = y0, the function g(τ ) in (1.23) is identically equal to zero, the
result in (3.56) follows immediately. In what follows, we shall derive the results in Cases (II)–(IV).
3.1. Case II: x = x0 6= 0 and y = y0. Now, the phase function simplifies to
g(τ ) = x2τ tanhτ 2,
which is real positive-valued and has one minimum at τ = 0. Therefore, we can use the Laplace’s method to derive the asymptotic expansion for (1.22). Note that g(τ ) and W (τ ) are even functions, hence
pt = 2 (2πt)3/2 Z ∞ 0 e−g(τ )/tW (τ )dτ. The series expansion of g and W are
(3.60) g(τ ) = x 2 2 τ 2− x 2 24τ 4+· · · = ∞ X k=0 akτk, W (τ ) = 1− 1 12τ 2+· · · = ∞ X k=0 bkτk.
In the notations of (2.42), m = 2, p = 4. By Laplace’s method pt∼ 2 (2πt)3/2 ∞ X s=0 Γs + 1 2 cst s+1 2 ∼(2π)23/2t ∞ X s=0 Γs + 1 2 csts/2, t → 0+,
where the coefficients cs can be calculated by Wojdylo’s formula in (2.48)-(2.49) with ak
and bk given in (3.60). A slight calculation shows
(3.61) c0 = √ 2 2|x|, c1 = 0, c2 = √ 2 24|x|3, c4 = −√2 320|x|5, c5 = 0.
3.2. Case III: x =−x0 6= 0 and |y − y0| = π2x2. Now,
(3.62) g(τ ) =−iτ(y − y0) + x2τ coth τ 2 = τ x 2h cothτ 2 − i π 2 i . It is readily seen that
g(τ ; x0, y0; x, y) = f (τ /2; 2x, 0, y− y0),
where f is the modified complex action function of the Heisenberg subLaplacian given in (1.18). Now g(τ ) has only one saddle point τ1 = iπ, which is also a branch point of W (τ ).
The saddle point corresponds to a generic geodesic, and the branch point corresponds to an exceptional geodesic, and these two geodesic coincide. The asymptoics of the heat kernel (1.22) can be derived in a similar method as that used in the Heisenberg case. The coalescing of a saddle point and a branch point is treated in [34, Ch. VII,§3]. The result in [23] also applies to this case. To this end, we first expand g(τ ) and W (τ ) at τ1 = iπ,
(3.63) g(τ )− g(τ1) = ∞ X s=2 as(τ − τ1)s, W (τ ) = ∞ X s=0 bs(τ − iπ)s−1/2, |τ − iπ| < π.
The first few coefficients are a0 = 1 2π 2, a 1 = 0, a2 = 1 2, a3 =− iπ 24, b0 = (−iπ)1/2, b1 = 1 2√π(−i) 3/2, b 2 = 1 8π2 − 1 12 (−iπ)1/2, · · ·
In the notations of [23], m = 2, p = 3,p and α = 12. Then, the asymptotic expansion of the heat kernel is given by
(3.64) pt(x0, y0; x, y)∼ e−g(τ1)/t (2πt)3/2 ∞ X s=0 Γs 2+ 1 4 Cst s 2+ 1 4 ∼ e −d2(q0,q)2t (2π)3/2t5/4 ∞ X s=0 Γs 2+ 1 4 Cst s 2 as t→ 0+. The coefficients C
sare calculated by the formulas in (2.47)-(2.49) with m = 2,
p = 3 and α = 1/2, and as, bs given in (3.63).
3.3. Case IV-1: x = x0 = 0 and y 6= y0. We assume y > y0 without loss of generality.
Now, the heat kernel in (1.22) becomes (3.65) pt(x0, y0; x, y) = 1 (2πt)32 Z ∞ −∞ eiτy−y0t τ sinh τ 12 dτ.
Note that, τk = ikπ, k = 1, 2, 3,· · · , are branch points of W (τ), and each of these
singu-larities corresponds to a pair of geodesics joining (0, y0) and (0, y). Since the contribution
of all the other singularities is exponentially small compared with the first one, we just concentrate on the first one τ1 = iπ. Recall the series expansion of W (τ ) given in (3.63).
By the change of variables
τ 7→ u : i(τ − iπ)y− y0 t = u
and deforming the integral path to the Hankel loop reduce the integral in (3.65) to (3.66) pt(x0, y0; x, y) = 1 (2πt)32 e−π(y−y0)t Z (0+) −∞ eu ∞ X s=0 −it y− y0 s+1/2 wsun−1/2du
To derive the asymptotic expansion, we recall the Hankel’s loop integral representation of the gamma function
(3.67) 1 Γ(−λ) = 1 2πi Z (0+) −∞ uλeudu; see [28] or [29, eq. (5.9.2)]. Thus,
(3.68) pt(x0, y0; x, y)∼ 1 (2πt)32 e−π(y−y0)t ∞ X s=0 −it y− y0 s−1/2 ws 2πt y− y0 1 Γ(12 − n) ∼ 1 2πte −π(y−y0)t ∞ X s=0 √ 2π Γ(12 − s) (−i)s−1/2w s (y− y0)s+1/2 ts ∼ 1 2πte −d2(q0,q)2t ∞ X s=0 √ 2π Γ(12 − s) (−i)s−1/2ws (y− y0)s+1/2 ts
as t → 0+. Here, we have made use the fact that π(y− y
0) = d2(q0; q)/2 with d(q0; q)
denoting the subRiemannian distance between the two points q0(0, y0) and q(0, y).
3.4. Case IV-2: x =−x0 6= 0 and |y − y0| > π2x2. In this case, there are finitely many
of geodesic connections, corresponding to τk = iηk with η1 = π < η2 ≤ · · · ≤ ηp. The
shortest geodesic is the exceptional geodesic corresponding to τ1 = iπ. From the integral
point of view, the main contribution comes from the first singularity of W (τ ), that is τ1 = iπ.
The asymptotic expansion of (1.22) can be derived in a same manner as in Case (III-1). Make the change of variables
(3.69) u =−g(τ )− g(τ1) t = 1 t ∞ X s=1 gs(τ − τ1)s, gs =− g(s)(τ 1) s! . By the inverse function theorem,
τ − τ1 = ∞
X
s=1
αstsus.
Differentiating the last equation with respect to u and using (3.63), one has W (τ )dτ du = ∞ X s=0 cst s+1 2 u s−1 2 .
A slight calculation yields
(3.70) c0 = b0√α1 = b0 (−g1)1/2 , c1 = 3b0α2+ 2b1α21 2√α1 .
Under the change of variables (3.69), the integral in (1.22) reduces to a Hankel’s loop integral (3.71) pt(x, y; x0, y0) = 1 (2π)3/2te −d2 2t Z (0+) −∞ eu ∞ X s=0 cst s 2u s−1 2 du,
where d2/2 = g(τ1) and d is the subRiemannian distance between the two points (x0, y0)
and (x, y). Using the integral representation in (3.67) and Watson’s lemma, one has (3.72) pt(x, y; x0, y0)∼ 1 2πte −d2(q0,q)2t ∞ X s=0 √ 2π Γ(1−s2 )cst s/2 , as t→ 0+. Indeed, all the terms with odd s vanish since 1/Γ(1−s
2 ) = 0.
3.5. Case IV-3: other cases. What left are the following cases: (a) x6= ±x0. Here, x0 or x can be zero;
(b) x =−x0 6= 0 and |y − y0| < π2x2;
(c) x = x0 6= 0.
The geodesics in Cases (a) and (b) are all generic geodesics. There might be exceptional geodesic(s) in case (c), but the shortest geodesic is generic. In all these three cases, the action function g(τ ) has finite number of critical points (saddle points) τk = iηk,
k = 1, 2,· · · , p, with 0 < η1 < π ≤ η2 <· · · ≤ ηp. Here, p is a positive integer. The first
and all the other saddle points contribute exponentially small and hence negligible. At the first saddle point τ1, g and W have the following Taylor expansions,
(3.73) g(τ )− g(τ1) = ∞ X s=2 as(τ − τ1)s, W (τ ) = ∞ X s=0 bs(τ − τ1)s.
Then, the method of saddle point yields
(3.74) pt(x, y; x0, y0)∼ e−d22t (2πt)3/2 ∞ X n=0 CnΓ n + 1 2 tn+12 ∼ e −d2 2t (2π)3/2t ∞ X n=0 CnΓ n + 1 2 tn2, as t → 0+. Here, d2/2 = g(τ
1) and d is the subRiemannian distance between the two
points (x0, y0) and (x, y). The coefficients of Cn can be obtained by the same manner as
in Section 2. The Cn’s can be computed by formulas in (2.47)-(2.49) with m = 2, p = 3,
α = 1 and as, bs given in (3.73).
3.6. Some observations. The heat kernel of the Grushin operator has the small-time asymtotics in the following form
pt(x0, y0; x, y)∼
C tαe
−d2 2t,
where C is a constant, d is the subRiemannian distance between the points q0(x0, y0) and
q(x, y), and 2α = 3 = Q > n, x = x0 = 0, y = y0, diagonal; 2 = Q = n, x = x0 6= 0, y = y0, diagonal; 5/2 = n + 1/2, x =−x0 6= 0, |y − y0| = πx2/2,
off-diagonal and cut-conjugate; 2 = n, off-diagonal and not cut-conjugate points.
Here, n = 2 is the topological dimension and Q is the Hausdorff dimension. We see that in most of the cases α = Q/2. The case of 2α = 5/2 corresponds to the situation when (x0, y0) and (x, y) are cut-conjugate points; see [2].
Case II two is the diagonal case, for which we have pt(x, y; x, y)∼ 1 2πt|x| 1 + 1 24|x|2t +· · · , as t → 0+.
The second term here is related to the scalar curvature in the sense that 24|x|1 2 =− R(x,y)
96 .
The scalar curvature R(x, y) is calculated in Appendix C.
4. Uniform asymptotic expansions, the discontinuity of α
4.1. Heisenberg subLaplacian. Let us consider the Heisenberg case first. We have shown that the leading power of t in the small-time asymptotic expansion for the heat kernel (1.17) varies as the point (x1, x2, y) varies. To be precise, as (x1, x2) approaches
(0, 0), the power α of t changes from 3/2 to 2; see eqs. (2.54)-(2.55). From the integral point of view, the discontinuity of α is due to the coalescing of the two saddle points τ1 and
θ2 both tend to π askxk → 0, and τ = iπ is also a simple pole of the phase function f(τ).
For such a case, Frenzen and Wong [14] derived a uniform asymptotic approximation in terms of Bessel function; see also [34, Ch. VII]. Their idea is to introduce a rational mapping τ 7→ u by
(4.75) − 2f(iη) = u − A
2(σ)
u ,
where σ = kxk2y2 and the function A(σ) is determined as follows. Note that
−2ifη(iη)
dτ
du = 1 + A(σ)2
u2 .
In order to have a one-to-one mapping in the region of interest, one requires dηdu 6= 0 or ∞. Note that fη(iη1) = 0 and fη(iη2) = 0, thus we let τ1 and τ2 correspond to u = iA(σ)
and u =−iA(σ), respectively. Therefore,
(4.76) A(σ) = if (iη1) = i
d2
2,
where d is the subRiemannian distance between (x, y) and the origin. Here, we have made use of the fact that f (iη1) = d2/2. By the transformation τ 7→ u, the heat kernel
(1.17) reduces to (4.77) pt(x, y) = 1 (2πt)3 Z u−1h(u) exp 1 t u−A(σ) 2 u du, where (4.78) h(u) = uV τ (u) dτ du
is analytic near u = 0. Recall Schl¨afli’s integral representation of the Bessel function Jν(z) (see [29, eq. (10.9.19)]) Jν(z) = 1 2πi z 2 νZ (0+) −∞ t−(ν+1)exp t− z 2 4t dt. The change of variables t = λu/2 leads to
(4.79) Jν(λz) = zν 2πi Z (0+) −∞ u−(ν+1)exp λ 2 u− z 2 u du. Comparing the last equation with (4.77) yields
pt(x, y)∼ 1 t2J0 A(σ) t a0(x, y) + O(t−1), as t→ 0+,
where a0(x, y) is a function of x, y. Follow the approach of Fenzen and Wong [14], we can
derive an asympttoic expansion of the form (4.80) pt(x1, x2, y)∼ 1 t2 ( J0(A(σ)/t) ∞ X s=0 asts+ J1(A(σ)/t) A(σ) ∞ X s=0 bsts+1, ) as t→ 0+.
4.2. Grushin operator. In the small-time asymptotics of the heat kernel of the Grushin operator, we see that the leading power of t, α, changes from 1 to 5/4 as q approaches the cut-conjugate point of q0. From the integral point of view, this discontinuity of α
is due to the coalescing of the saddle point τ1 and the singularity of W (τ ) at τ = iπ.
The treatment of such coalesce of critical points can be found in Wong [34, Ch. VII, §3], and the uniform asymptotic expansion involves parabolic cylinder functions. The discontinuous change of α from 3/2 to 1 can be smoothed out in a same manner as in the Heisenberg case. Since there is no alternation in the methods, we omit the details here.
5. Discussions
We have derived small-time asymptotic expansions for the heat kernels of the Heisen-berg subLaplacian and the Grushin operator. Our derivation is based on the integral representations, and 1/t serves as a large parameter as t → 0+. The techniques in de-riving asymptotics of integrals, such as the methods of stationary phase and steepest descent, are the main tools here. The leading power of t in the expansions of pt(q0; q)
is different for different situations of q0 and q. This discontinuity of the power α can be
smoothed out by the uniform expansion techniques. As all the known subelliptic heat kernels have Laplace-type integral representations, the techniques described here can also be applied to such cases. The two examples we take here are both subelliptic operators on non-compact manifold. On the other hand, elliptic heat kernel asymptotics show that the compact cases pose more interesting properties, since their eigenvalues are discrete and one can investigate the trace asymptotic expansions. In the future work, we shall consider subelliptic heat kernel asymptotics on compact manifolds, such as the subLaplacians on the odd-dimension spheres S2n+1 considered in [16].
Appendix A. SubRiemannian geometry
The term bracket-generating condition is the following condition on the vector fields {Xj}kj=1, or equivalently, on the distribution D := span {Xj}kj=1. Recall that the
Lie bracket of two vector fields Xi and Xj is [Xi, Xj] = XiXj − XjXi. One can define
a sequence of distributions {Ds}∞
s=1 recursively by D1 = D and Ds+1 = D ∪ [D, Ds] for
s = 1, 2,· · · . Note that Ds ⊂ T M for each s ≥ 1, here T M denotes the tangent bundle
of M . If there exists a positive integer S, such that DS = T M , then we say the vector
fields {Xj}kj=1 are bracket-generating. The smallest integer S such that DS = T M is
called the step of these vector fields.
Assume{Xj}kj=1are bracket-generating vector fields on the n-dimensional manifold M .
Since k < n, we have several directions missing. However, Chow’s theorem [13] showed that one can define a nature metric on M . To start, consider a curve γ(s) : [0, 1] → M whose tangent is in D = span {X1,· · · , Xk} and given by
γ0(s) =
k
X
j=1
aj(s)Xj.
By assuming {Xj}kj=1 are orthonormal, one can define the length
`(γ) = Z 1
0
p
Such curves with tangents being in D are called horizontal curves or admissible curves. Chow’s theorem says the following: if the vector fields {Xj}kj=1 are
bracket-generating, then any two points q0, q ∈ M can be joined by at least one horizontal curve.
By minimizing all the lengths of the horizontal curves joining these two points, one can define the distance between q0 and q. This distance d(q0, q) is known as the
Carnot-Carath´eodory distance (or subRiemannian distance). Therefore, Chow’s theorem shows that one can define a metric structure g on M , but this is not a Riemannian metric. The metric space (M,D, g) is called subRiemannian geometry or Carnot-Carath´eodory space.
Despite of the similarity in the names, subRiemannian geometries are very different from Riemannian ones. For instance, the geodesic structures are different, the local uniqueness of the shortest geodesics for the Riemannian geometries is no longer valid in subRiemannian geometries. Another difference is the Hausdorff dimension. The Haus-dorff dimension of the Riemannian geometry equals to n, however the HaussHaus-dorff dimen-sion of subRiemannian geometry is larger than n. Mitchell [25] showed that the Hausdorff dimension of (M,D, g) can be calculated as
(A.81) Q =
S
X
s=1
(dimDs− dim Ds−1)s, with dimD0 = 0.
Appendix B. Cut points and conjugate points
Fix a point q0 ∈ M, a point q is a conjugate point of q0if there exists a nonzero Jacobi
fields J along one geodesic joining q0 and q such that J (q0) =J (q) = 0. An equivalent
definition of the conjugate point is by the exponential map, see for example, [1]. If q is a conjugate point of q0 along a shortest geodesic γ, this geodesic will not be shortest any
more after this point q. This indicates that the distance d(q0, q) as a function of q is not
smooth at a conjugate point, and the loss of smoothness of d(q0, q) explains why the heat
kernel asymptotic behavior changes at a conjugate point.
Take the Grushin plane as an example. Given a fixed point q0(x0, y0) with x0 6= 0, a
point q(x, y) is a conjugate point of q0 if and only if |y − y0| = kπ2 x2 and x = (−1)kx0
for some positive integer k. However, the small-time asymptotic expansion for the heat kernel changes only at the first two conjugate points corresponding to k = 1. The explanation is as follows. The first two conjugate points here are also cut points, that is q ∈ Cut(q0)∪ Conj(q0). Here, Cut(q0) and Conj(q0) denote the cut locus (the set of all
cut points) and the conjugate locus (the set of all conjugate points) of q0, respectively.
To be clear, we recall that q is a cut point of q0 if: (i) there are more than one shortest
geodesic joining q and q0; or (ii) q is a conjugate point of q0 along one shortest geodesic.
Therefore, for a given point q0, there might be points are cut points but not conjugate
points, and vice versa. Take the Grushin plane as an example for illustration. The cut locus of q0(0, y0) is {(0, y); y 6= y0}, but these points are not conjugate points. For the
point q0(x0, y0) with x0 6= 0, the conjugate points are given above as {(x, y); |y − y0| = kπ
2 x
2, x = (−1)kx
0, k = 1, 2, 3,· · · }, but only the first two conjugate points associated
with k = 1 are cut points, all the others are not in the cut locus of q0. Our result here
confirms the observation of Barilari, Boscain and Neel [2], that the leading power of t in the subelliptic heat kernel asymptotics changes at the point q ∈ Conj(q0)∪ Cut(q0).
This fact is also noticed by Molchanov [26] for Riemannian case. How about the other conjugate points q∈ Conj(q0)\ Cut(q0)? This happens when q is conjugate along a
points will change the behavior of exponentially small terms, which can not be seen from the asymptotic expansion we consider here.
x y −1 q0 1 q1 q−1 q2 q−2 q3 q−3 q4 q−4 π/2 π 3π/2 x y O −2π 1.28π
Figure 2. Cut points and conjugate points of q0 in the Grushin plane.
(a) q0 = (−1, 0): Conj(q0) = {q±1, . . . q±k, . . .}, these points are conjugate
with q0 along the geodesic plotted in the brown curve; Cut(q0) = {q1, q−1},
so q1 and q−1 are the only cut-conjugate points of q0.
(b) q0 = (0, 0): Cut(q0) = {(0, y); y 6= 0} and Conj(q0) = Ø, any point
q(0, y) is joined with q0 by two shortest geodesics plotted in a solid and a
dotted curves, illustration is done for q = (0,−2π) and q = (0, 1.28π).
Appendix C. Scalar curvature
Recall that the scalar curvature is the trace of the Ricci curvature tensor
(C.82) R =X
i,j
gijRij,
where the Ricci tensor can be calculated via the Christoffel symbols
(C.83) Rij = X ` " ∂Γ`ij ∂x` − ∂Γ` i` ∂xj + X m ΓmijΓ``m− Γmi`Γ`jm # and (C.84) Γmij = 1 2 X k gmk ∂gki ∂xj + ∂gkj ∂xi − ∂gij ∂xk .
Consider the Grushin vector fields in (1.20) for m = 1, and the coordinates (x, y) are understood as (x1, x2) in the above formulas. The metric is given by
(C.85) (gij) = 1 0 0 x−2 , (gij) = (gij)−1= 1 0 0 x2 . A slight calculation shows
(C.86) R(x, y) =− 4
x2, x6= 0.
Acknowledgement
This research project was initiated when the authors visited the National Center for Theoretical Sciences (NCTS), Hsinchu, Taiwan during January 2013, and the final ver-sion of the paper was completed while the authors visited NCTS during July 2014. They would like to express their profound gratitude to the Director of NCTS, Professor Winnie Li, for her invitation and for the warm hospitality extended to them during their stay in Taiwan. The work of D.-C. Chang is partially supported by Hong Kong RGC compet-itive earmarked research grants #600607, #601410 and a competcompet-itive research grant at Georgetown University. The work of Y. Li is partially supported by a General Research Fund from Hong Kong Research Grant Committee (Grant no. 201513).
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Department of Mathematics and Department of Computer Sciences, Georgetown Uni-versity, Washington D.C. 20057, USA, and Department of Mathematics, Fu Jen Catholic University, Taipei 242, Taiwan, ROC
E-mail address: [email protected]
Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong E-mail address: [email protected]
