The Subelliptic Heat Kernel on the CR sphere
aa r X i v : . [ m a t h . A P ] D ec The Subelliptic Heat Kernel on the CR sphere
Fabrice Baudoin ∗ , Jing WangDepartment of Mathematics, Purdue UniversityWest Lafayette, IN, USA Abstract
We study the heat kernel of the sub-Laplacian L on the CR sphere S n +1 . Anexplicit and geometrically meaningful formula for the heat kernel is obtained. Asa by-product we recover in a simple way the Green function of the conformal sub-Laplacian − L + n that was obtained by Geller [12], and also get an explicit formulafor the sub-Riemannian distance. The key point is to work in a set of coordinates thatreflects the symmetries coming from the fibration S n +1 → CP n . Contents S n +1 S n +1 ∗ First author supported in part by NSF Grant DMS 0907326 Introduction
The purpose of this work is to study the heat kernel of the sub-Laplacian of the stan-dard CR structure on S n +1 . More precisely, we will be interested in explicit analyticrepresentations and small time asymptotics of the kernel.A key point in our study is to take advantage of the radial symmetries of the fibration S n +1 → CP n to introduce coordinates that are adapted to the geometry of the problem.In particular, it is shown that the kernel has a cylindric invariance property and actuallyonly depends on two variables r, θ . The variable θ is the local fiber coordinate of thefibration and r is a radial coordinate on CP n . In these coordinates the cylindric part ofthe sub-Laplacian L is the operator˜ L = ∂ ∂r + ((2 n −
1) cot r − tan r ) ∂∂r + tan r ∂ ∂θ . Eigenvalues and eigenvectors for ˜ L are computed and as a consequence of the generalMinakshisundaram-Pleijel expansion theorem we deduce that the subelliptic heat kernel(issued from the north pole) can be written p t ( r, θ ) = Γ( n )2 π n +1 + ∞ X k = −∞ + ∞ X m =0 (2 m + | k | + n ) (cid:18) m + | k | + n − n − (cid:19) e − λ m,k t + ikθ (cos r ) | k | P n − , | k | m (cos 2 r ) , where P n − , | k | m is a Jacobi polynomial and λ m,k = 4 m ( m + | k | + n ) + 2 | k | n . This formulais very useful to study the long-time behavior of the heat kernel but seems difficult touse in the study of small-time asymptotics or for the purpose of proving upper and lowerbounds. In order to derive small-time asymptotics of the kernel, we give another analyticexpression for p t ( r, θ ) which is much more geometrically meaningful. This formula isobtained thanks to the observation that the Reeb vector field T of the CR structure of S n +1 commutes with the sub-Laplacian. This commutation implies that the subellipticheat semigroup e tL can be written e − tT e t ∆ , where ∆ is the Laplace-Beltrami operator ofthe standard Riemannian structure on S n +1 . The formula we obtain is explicit enoughto recover Geller’s formula (see [12]) for the fundamental solution of the conformal sub-Laplacian − L + n . By using the steepest descent method, it also allows to derive thesmall-time asymptotics of the heat kernel. A by-product of this small-time asymptotics isa previously unknown explicit formula for the sub-Riemannian distance.To put things in perspective let us observe that the study of explicit expressions forsubelliptic heat kernels has generated a great amount of work (see [1], [2] [4], [3], [5][7], [11] and the references therein). The motivations for finding explicit formulas arenumerous, among them we can cite: sharp constant in functional inequalities (see [6],[14]), computation of the sub-Riemannian metric (see [3]), sharp upper and lower boundsfor the heat kernel (see [5],[10]), and semigroup sub-commutations (see [13]). However,despite such numerous works, very few explicit and tractable formulas are actually knownand most of them are restricted to a Lie group framework. The present work gives explicitand tractable expressions that hold in a natural sub-Riemannian model.2 The sub-Laplacian on S n +1 We consider the odd dimensional sphere S n +1 = { z = ( z , · · · , z n +1 ) ∈ C n +1 , k z k = 1 } . It is a strictly pseudo convex CR manifold (see [9]) whose geometry can be described asfollows. There is a natural group action of S on S n +1 which is defined by( z , · · · , z n ) → ( e iθ z , · · · , e iθ z n ) . The generator of this action shall be denoted by T throughout the paper. We have for all f ∈ C ∞ ( S n +1 ) T f ( z ) = ddθ f ( e iθ z ) | θ =0 , so that T = i n +1 X j =1 (cid:18) z j ∂∂z j − z j ∂∂z j (cid:19) . This action induces a fibration (circle bundle) from S n +1 to the projective complex space CP n . The vector field T is the Reeb vector field (characteristic direction) of the pseudo-Hermitian contact form η = − i n +1 X j =1 ( z j dz j − z j dz j ) . For j = 1 , · · · , n + 1, let us denote T j = ∂∂z j − z j S , where S = P n +1 k =1 z k ∂∂z k , and define the second order differential operator L on C ∞ ( S n +1 )as follows: L = 2 n +1 X j =1 ( T j T j + T j T j ) . (2.1)It can be checked that L is the CR sub-Laplacian of the previously described structure(see for instance [6], [8]). It is essentially self-adjoint on C ∞ ( S n +1 ) with respect to theuniform measure of S n +1 and related to the Laplace-Beltrami operator ∆ of the standardRiemannian structure on S n +1 by the formula: L = ∆ − T .
3e can observe, a fact which will be important for us, that L and T commute, that is, onsmooth functions T L = LT .To study L we now introduce a set of coordinates that takes into account the sym-metries of the fibration S n +1 → CP n . Let ( w , · · · , w n , θ ) be local coordinates for S n +1 , where ( w , · · · , w n ) are the local inhomogeneous coordinates for CP n given by w j = z j /z n +1 , and θ is the local fiber coordinate. i.e., ( w , · · · , w n ) parametrizes thecomplex lines passing through the north pole , while θ determines a point on the line thatis of unit distance from the north pole. More explicitly, these coordinates are given by( w , · · · , w n , θ ) −→ w e iθ p ρ , · · · , w n e iθ p ρ , e iθ p ρ ! , (2.2)where ρ = qP nj =1 | w j | , θ ∈ R / π Z , and w ∈ CP n . In these coordinates, it is clear that T = ∂∂θ . Our goal is now to compute the sub-Laplacian L . In the sequel we denote R = n X j =1 w j ∂∂w j . Proposition 2.1
In the coordinates (2.2), we have L = 4(1 + ρ ) n X k =1 ∂ ∂w k ∂w k + 4(1 + ρ ) RR + ρ ∂ ∂θ − i ( ρ + 1)( R − R ) ∂∂θ . Proof . From (2.1), we know that L = 2 n +1 X k =1 (cid:18) ∂ ∂z k ∂z k + ∂ ∂z k ∂z k (cid:19) − n +1 X k,j =1 (cid:18) z j z k ∂ ∂z j ∂z k + z j z k ∂ ∂z j ∂z k (cid:19) − n ( S + S ) . By using now the diffeomorphism( w , · · · , w n , θ, κ ) −→ κw e iθ p ρ , · · · , κw n e iθ p ρ , κe iθ p ρ ! , and then restrict to the sphere on which we have: κ = 1 , ∂∂κ = 0 , we compute that on S n +1 , for 1 ≤ k ≤ n∂∂z k = p ρ e − iθ ∂∂w k ∂∂z k = p ρ e iθ ∂∂w k We call north pole the point with complex coordinates z = 0 , · · · , z n +1 = 1, it is therefore the pointwith real coordinates (0 , · · · , , , ∂∂z n +1 = − p ρ e − iθ n X j =1 w j ∂∂w j − i ∂∂θ ∂∂z n +1 = − p ρ e iθ n X j =1 w j ∂∂w j + 12 i ∂∂θ . Tedious but straightforward computations lead then to n +1 X k,j =1 z j z k ∂ ∂z j ∂z k = 14 ∂ ∂θ − i ∂∂θ which implies that n +1 X k,j =1 (cid:18) z j z k ∂ ∂z j ∂z k + z j z k ∂ ∂z j ∂z k (cid:19) = 12 ∂ ∂θ . Moreover, it is not hard to compute n +1 X k =1 (cid:18) ∂ ∂z k ∂z k + ∂ ∂z k ∂z k (cid:19) = 2(1 + ρ ) (cid:18) ∂ ∂w k ∂w k + RR + 14 ∂ ∂θ + 12 i ( R − R ) ∂∂θ (cid:19) , where R = P nk =1 w k ∂∂w k . Finally, it is easy to see that S + S = 0. Hence we have theconclusion. (cid:3) Remark 2.2
Notice that T = ∂∂θ , and thus the Laplace-Beltrami operator is given by: ∆ = L + ∂ ∂θ = 4(1 + ρ ) n X k =1 ∂ ∂w k ∂w k + 4(1 + ρ ) RR + (1 + ρ ) ∂ ∂θ − i ( ρ + 1)( R − R ) ∂∂θ Due to the symmetries of the fibration S n +1 → CP n , in the study of the heat kernel,it will be enough to compute the radial part of L with respect to the cylindrical variables( ρ, θ ).Let us consider the following second order differential operator˜ L = (cid:0) ρ (cid:1) ∂ ∂ρ + (cid:18) (2 n − ρ ) ρ + (1 + ρ ) ρ (cid:19) ∂∂ρ + ρ ∂ ∂θ , which is defined on the space D of smooth functions f : R ≥ × R / π Z → R that satisfies ∂f∂ρ = 0 if ρ = 0. It is seen that ˜ L is essentially self-adjoint on D with respect to themeasure ρ n − (1+ ρ ) n + 12 dρdθ . 5 roposition 2.3 Let us denote by ψ the map from S n +1 to R ≥ × R / π Z such that ψ w e iθ p ρ , · · · , w n e iθ p ρ , e iθ p ρ ! = ( ρ, θ ) . For every f ∈ D , we have L ( f ◦ ψ ) = ( ˜ Lf ) ◦ ψ. Proof . Notice that by symmetries of the fibration, we have n X k =1 ∂ ∂w k ∂w k ! ( f ◦ ψ ) = (cid:18)(cid:18) ∂ ∂ρ + 2 n − ρ ∂∂ρ (cid:19) f (cid:19) ◦ ψ and R ( f ◦ ψ ) = R ( f ◦ ψ ) = (cid:18)(cid:18) ρ ∂∂ρ (cid:19) f (cid:19) ◦ ψ. Together with Proposition 2.1, we have the conclusion. (cid:3)
Finally, instead of ρ , it will be expedient to introduce the variable r which is definedby ρ = tan r . It is then easy to see that we can write ˜ L as˜ L = ∂ ∂r + ((2 n −
1) cot r − tan r ) ∂∂r + tan r ∂ ∂θ . (2.3)This is the expression of ˜ L which shall be the most convenient for us and that is going tobe used throughout the paper.We can observe that ˜ L is symmetric with respect to the measure dµ r = 2 π n Γ( n ) (sin r ) n − cos rdrdθ. The normalization is chosen in such a way that Z π − π Z π dµ r = µ ( S n +1 ) = 2 π n +1 Γ( n + 1) . Remark 2.4
In the case of S ( n = 1) , which is isomorphic to the Lie group SU (2) , weobtain ˜ L = (1 + ρ ) ∂ ∂ρ + (cid:18) (1 + ρ ) ρ (cid:19) ∂∂ρ + ρ ∂ ∂θ = ∂ ∂r + 2 cot 2 r ∂∂r + tan r ∂ ∂θ This coincides with the result in [3]. emark 2.5 By Remark 2.2, we see that the radial part of the Laplace-Beltrami operatorin cylindrical coordinates is ∆ r = ∂ ∂r + ((2 n −
1) cot r − tan r ) ∂∂r + 1cos r ∂ ∂θ . On the other hand, since in the coordinates (2.2), we have z n +1 = cos re iθ , it is clear thatthe Riemannian distance form the north pole δ , satisfies cos δ = cos r cos θ. An easy calculation shows that by making the change of variable cos δ = cos r cos θ , theoperator ∆ r acts on functions depending only on δ as ∂ ∂δ + 2 n cot δ ∂∂δ . This expression is known to indeed be the expression of the radial part of ∆ in sphericalcoordinates. S n +1 From the expression of L above, it is not hard to see that the kernel of P t = e tL issuedfrom the north pole only comes from the radial part ˜ L and depends on ( r, θ ). We denoteit by p t ( r, θ ). Proposition 3.1
For t > , r ∈ [0 , π ) , θ ∈ [ − π, π ] , the subelliptic kernel has the followingspectral decomposition: p t ( r, θ ) = Γ( n )2 π n +1 + ∞ X k = −∞ + ∞ X m =0 (2 m + | k | + n ) (cid:18) m + | k | + n − n − (cid:19) e − λ m,k t + ikθ (cos r ) | k | P n − , | k | m (cos 2 r ) , where λ m,k = 4 m ( m + | k | + n ) + 2 | k | n and P n − , | k | m ( x ) = ( − m m m !(1 − x ) n − (1 + x ) | k | d m dx m ((1 − x ) n − m (1 + x ) | k | + m ) is a Jacobi polynomial.Proof . The idea is to expand p t ( r, θ ) as a Fourier series in θ . Let p t ( r, θ ) = + ∞ X k = −∞ e ikθ φ k ( t, r ) ,
7e this Fourier expansion. Since p t satisfies ∂p t ∂t = Lp t , we have ∂φ k ∂t = ∂ φ k ∂r + ((2 n −
1) cot r − tan r ) ∂φ k ∂r − k tan rφ k . By writing φ k ( t, r ) in the form φ k ( t, r ) = e − n | k | t (cos r ) | k | g k ( t, cos 2 r ) , we get ∂g k ∂t = 4Ψ k ( g k ) , (3.4)where Ψ k = (1 − x ) ∂ ∂x + [( | k | + 1 − n ) − ( | k | + 1 + n ) x ] ∂∂x . In fact (3.4) is well-known as the Jacobi differential equation, and the eigenvectors aregiven by P n − , | k | m ( x ) = ( − m m m !(1 − x ) n − (1 + x ) | k | d m dx m ((1 − x ) n − m (1 + x ) | k | + m ) , which satisfies that Ψ k ( P n − , | k | m )( x ) = − m ( m + n + | k | ) P n − , | k | m ( x ) . Therefore, we obtain the spectral decomposition p t ( r, θ ) = + ∞ X k = −∞ + ∞ X m =0 α m,k e − [4 m ( m + | k | + n )+2 | k | n ] t e ikθ (cos r ) | k | P n − , | k | m (cos 2 r ) , where the constants α m,k ’s are to be determined by the initial condition at time t = 0.To compute them, we use the fact that ( P n − , | k | m ( x )(1 + x ) | k | / ) m ≥ is an orthogonalbasis of L ([ − , , (1 − x ) n − dx ), i.e., Z − P n − , | k | m ( x ) P n − , | k | l ( x )(1 − x ) n − (1+ x ) | k | dx = 2 n + | k | m + | k | + n Γ( m + n )Γ( m + | k | + 1)Γ( m + 1)Γ( m + n + | k | ) δ ml . For a smooth function f ( r, θ ), we can write f ( r, θ ) = + ∞ X k = −∞ + ∞ X m =0 b k,m e ikθ P n − , | k | m (cos 2 r ) · (1 + cos 2 r ) | k | / where the { b k,m } ’s are constants, and thus f (0 ,
0) = + ∞ X k = −∞ + ∞ X m =0 b k,m P n − , | k | m (1)2 | k | / . Z π − π Z π p t ( r, θ ) f ( r, θ ) dµ r = 2 π n Γ( n ) Z π − π Z π p t ( r, θ ) f ( r, θ )(sin r ) n − cos rdrdθ = 2 π n Γ( n ) + ∞ X k = −∞ + ∞ X m =0 Z π − π Z π α m,k b m,k | k | / e − λ m,k t | P n − , | k | m | (cos r ) | k | +1 (sin r ) n − drdθ = 2 π n Γ( n ) + ∞ X k = −∞ + ∞ X m =0 α m,k b m,k e − λ m,k t − n −| k | / − Z π − π Z π n + | k | +1 (cos r ) | k | +1 | P n − , | k | m | (sin r ) n − dr ! dθ = 2 π n Γ( n ) + ∞ X k = −∞ + ∞ X m =0 α m,k b m,k e − λ m,k t − n −| k | / − (2 π ) || P n − , | k | m || where λ m,k = 4 m ( m + | k | + n ) + 2 | k | n , we obtain thatlim t → Z π − π Z π p t f dµ r = f (0 , α m,k = Γ( n )2 π n +1 (2 m + | k | + n ) (cid:0) m + | k | + n − n − (cid:1) . (cid:3) The spectral decomposition of the heat kernel is explicit and useful but is not reallygeometrically meaningful. We shall now study another representation of the kernel whichis more geometrically meaningful and which will turn out to be much more convenientwhen dealing with the small-time asymptotics problem. The key idea is to observe thatsince ∆ and ∂∂θ commutes, by Remark 2.2 we formally have e tL = e − t ∂ ∂θ e t ∆ . (3.5)This gives a way to express the sub-Riemannian heat kernel in terms of the Riemannianone. Let us recall that the Riemannian heat kernel writes q t (cos δ ) = Γ( n )2 π n +1 + ∞ X m =0 ( m + n ) e − m ( m +2 n ) t C nm (cos δ ) , (3.6)where, as above, δ is the Riemannian distance from the north pole and C nm ( x ) = ( − m m Γ( m + 2 n )Γ( n + 1 / n )Γ( m + 1)Γ( n + m + 1 /
2) 1(1 − x ) n − / d m dx m (1 − x ) n + m − / , is a Gegenbauer polynomial. Another expression of q t (cos δ ) which is useful for the com-putation of small-time asymptotics is q t (cos δ ) = e n t (cid:18) − π sin δ ∂∂δ (cid:19) n V (3.7)9here V ( t, δ ) = √ πt P k ∈ Z e − ( δ − kπ )24 t .Using the commutation (3.5) and the formula cos δ = cos r cos θ , we then infer thefollowing proposition. Proposition 3.2
For t > , r ∈ [0 , π/ , θ ∈ [ − π, π ] , p t ( r, θ ) = 1 √ πt Z + ∞−∞ e − ( y + iθ )24 t q t (cos r cosh y ) dy. (3.8) Proof . Let h t ( r, θ ) = 1 √ πt Z + ∞−∞ e − ( y + iθ )24 t q t (cos r cosh y ) dy, and ˜ L = ∂ ∂r + ((2 n −
1) cot r − tan r ) ∂∂r , then we have˜ L = ˜ L + tan r ∂ ∂θ . Using the fact that ∂∂t e − y t √ πt = ∂ ∂y e − y t √ πt and ∂∂t ( q t (cos r cos θ )) = ∆( q t (cos r cos θ )) , we get˜ Lh t ( r, θ ) = (cid:18) ˜ L + tan r ∂ ∂θ (cid:19) h t ( r, θ )= Z + ∞−∞ ˜ L e − ( y + iθ )24 t √ πt q t (cos r cosh y ) + tan r ∂ ∂θ e − ( y + iθ )24 t √ πt q t (cos r cosh y ) dy = Z + ∞−∞ (cid:18) ∆ − r ∂ ∂θ (cid:19) e − ( y + iθ )24 t √ πt q t (cos r cosh y ) + tan r ∂ ∂θ e − ( y + iθ )24 t √ πt q t (cos r cosh y ) dy = Z + ∞−∞ e − ( y + iθ )24 t √ πt ∂q t ∂t + 1cos r ∂∂t e − ( y + iθ )24 t √ πt q t − tan r ∂∂t e − ( y + iθ )24 t √ πt q t dy = ∂∂t h t ( r, θ )On the other hand, it suffices to check the initial condition for functions of the form f ( r, θ ) = e iλθ g ( r ) where λ ∈ R and g is smooth. We observe that Z π Z π h t ( r, θ ) f ( r, θ ) dµ r = e tλ ( e t ∆ r g )(0) . Thus h t ( r, θ ) is the desired subelliptic heat kernel. (cid:3) roposition 3.3 For λ ∈ C , Re λ > , r ∈ [0 , π/ , θ ∈ [ − π, π ] , Z + ∞ p t ( r, θ ) e − n t − λt dt = Z ∞−∞ Γ( n ) dy n +2 π n +1 (cid:16) cosh p y + 4 λ − cos r cos( θ + iy ) (cid:17) n (3.9) Proof . Since Z + ∞ p t ( r, θ ) e − n t − λt dt = Z + ∞ e − λt p t ( r, θ ) e − n t dt = 1 √ π Z + ∞−∞ Z + ∞ e − n t − y λ t q t (cos r cos( θ + iy )) dt √ t dy We want to compute Z + ∞ e − n t − y λ t q t (cos r cos( θ + iy )) dt √ t . Notice that Z + ∞ e − n t − y λ t e t ∆ dt √ t = Z + ∞ e − y λ t e − t ( n − ∆) dt √ t = √ π √ n − ∆ e − √ y +4 λ √ n − ∆ However, by the result in Taylor [[16], pp. 95], we have that A − e − tA = 12 π − ( n +1) Γ( n )(2 cosh t − δ ) − n , where A = √ n − ∆ and cos δ = cos r cos( θ + iy ). Plug in t = p y + 4 λ , we obtain1 √ n − ∆ e − √ y +4 λ √ n − ∆ = Γ( n )2 n +1 π n +1 (cid:16) cosh p y + 4 λ − cos r cos( θ + iy ) (cid:17) n hence complete the proof. (cid:3) We can deduce the Green function of − L + n immediately from the above proposition. Proposition 3.4
The Green function of the conformal sub-Laplacian − L + n is given by G ( r, θ ) = Γ (cid:0) n (cid:1) π n +1 (1 − r cos θ + cos r ) n/ roof . Let us assume r = 0, θ = 0, and let λ → G ( r, θ ) = Γ( n )2 n +2 π n +1 Z + ∞−∞ dy (cosh y − cos r cos( θ + iy )) n = Γ( n )2 n +2 π n +1 Z + ∞−∞ dy (cosh y (1 − cos r cos θ ) − i cos r sin θ sinh y ) n = Γ( n )2 n +2 π n +1 − r cos θ + cos r ) n/ Z + ∞−∞ dy (cosh y ) n Notice that Z + ∞−∞ dy (cosh y ) n = π ( n − n − n > n − n − n > n !! denotes the double factorial such that n !! = n · ( n − · · · · · n > n · ( n − · · · · · n > n ) ( n − n − n − π Γ (cid:16) n (cid:17) when n > n − Γ (cid:16) n (cid:17) when n > n ) Z + ∞−∞ dy (cosh y ) n = 2 n − Γ (cid:16) n (cid:17) for all n ∈ Z > This implies our conclusion. (cid:3)
Remark 3.5
This agrees with the result by Geller in [12].
First, we study the asymptotics of the subelliptic heat kernel when t → q t (cos δ ) = 1(4 πt ) n + (cid:18) δ sin δ (cid:19) n e − δ t (cid:18) (cid:18) n − n ( n − δ − δ cos δ ) δ sin δ (cid:19) t + O ( t ) (cid:19) , (3.10)where δ ∈ [0 , π ). Here δ is the Riemannian distance. Together with (3.8), we first deducethe following small-time-asymptotics of the subelliptic heat kernel on the diagonal.12 roposition 3.6 When t → , p t (0 ,
0) = 1(4 πt ) n +1 ( A n + B n t + O ( t )) , where A n = R ∞−∞ y n (sinh y ) n dy and B n = R ∞−∞ y n (sinh y ) n (cid:16) (cid:16) n + n ( n − y − y cosh y ) y sinh y (cid:17)(cid:17) dy .Proof . We know that p t (0 ,
0) = 1 √ πt Z ∞−∞ e − y t q t (cosh y ) dy. Plug in (3.10), we have the desired small time asymptotics. (cid:3)
Proposition 3.7
For θ ∈ (0 , π ) , t → , p t (0 , θ ) = θ n − n t n ( n − e − πθ − θ t (1 + O ( t )) Proof . Let θ ∈ (0 , π ), we have p t (0 , θ ) = 1 √ πt Z ∞−∞ e − y t q t (cosh( y − iθ )) dy By Cauchy integral theorem, this is the same as integrating along the horizontal line inthe complex plane by shifting up iθ from the real axis. i.e., p t (0 , θ ) = 1 √ πt Z ∞−∞ e − ( y + iθ )24 t q t (cosh y ) dy Moreover, by (3.10), we know that q t (cosh y ) ∼ t → πt ) n + (cid:18) y sinh y (cid:19) n e y t . This gives p t (0 , θ ) ∼ t → e θ t (4 πt ) n +1 Z ∞−∞ y n (sinh y ) n e − iyθ t dy By the residue theorem, we get Z ∞−∞ y n (sinh y ) n e − iyθ t dy = − πi X k ∈ Z + Res e − iyθ t y n (sinh y ) n , − kπi ! = − πi X k ∈ Z + n − ∂ n − ∂y n − " e − iyθ t n y n ( y + kπi ) n ( e y − e − y ) n y = − kπi W ( y ) = ( y + kπi ) n ( e y − e − y ) n , W ( y ) is analytic around − kπi , and satisfies W ( − kπi ) = 1( − kn n . Hence the residue is 1( n − ∂ n − ∂y n − h e − iyθ t n y n W ( y ) i y = − kπi . This is a product of e − iyθ t and a polynomial of degree n − /t . We are only interested inthe leading term which plays the dominant role when t →
0. Thus we have the equivalence − πi ( n − ∂ n − ∂y n − h e − iyθ t n y n W ( y ) i y = − kπi ∼ t → ( − kn + n π n +1 θ n − ( n − n − t n − e − kπθ t At the end, we conclude p t (0 , θ ) ∼ t → e θ t (4 πt ) n +1 X k ∈ Z + ( − kn + n π n +1 θ n − ( n − n − t n − e − kπθ t , that is p t (0 , θ ) = θ n − n t n ( n − e − πθ − θ t (1 + O ( t )) (cid:3) Come to the points that do not lie on the cut-locus, i.e., r = 0. First we deduce thecase for ( r, Proposition 3.8
For r ∈ (0 , π ) , we have p t ( r, ∼ e − r t (4 πt ) n + (cid:16) r sin r (cid:17) n r − r cot r as t → .Proof . By proposition 3.2, p t ( r,
0) = 1 √ πt Z ∞−∞ e − y t q t (cos r cosh y ) dy, together with (3.10), it gives that p t ( r, ∼ t → πt ) n +1 ( J ( t ) + J ( t ))14here J ( t ) = Z cosh y ≤ r e − y r cosh y ))24 t arccos(cos r cosh y ) p − cos r cosh y ! n dy and J ( t ) = Z cosh y ≥ r e − y − (cosh − r cosh y ))24 t cosh − (cos r cosh y ) p cos r cosh y − ! n dy We can analyze J ( t ) and J ( t ) by Laplace method. First, notice that in (cid:2) − cosh − ( r ) , cosh − ( r ) (cid:3) , f ( y ) = y + (arccos(cos r cosh y )) has a unique minimum at y = 0, where f ′′ (0) = 2(1 − r cot r ) . Hence by Laplace method, we can easily obtain that J ( t ) ∼ t → e − r t (cid:16) r sin r (cid:17) n r πt − r cot r On the other hand, on (cid:0) −∞ , − cosh − ( r ) (cid:1) ∪ (cid:0) cosh − ( r ) , + ∞ (cid:1) , the function g ( y ) = y − (cosh − (cos r cosh y )) has no minimum, which implies that J ( t ) is negligible with respect to J ( t ) in small t .Hence the conclusion. (cid:3) We can now extend the result to the θ = 0 case by applying the steepest descentmethod. Lemma 3.9
For r ∈ (0 , π ) , θ ∈ [ − π, π ] , f ( y ) = ( y + iθ ) + (arccos(cos r cosh y )) defined on the strip | Re ( y ) | < cosh − (cid:0) r (cid:1) has a critical point at iϕ ( r, θ ) , where ϕ ( r, θ ) is the unique solution in [ − π, π ] to the equation ϕ ( r, θ ) + θ = cos r sin ϕ ( r, θ ) arccos(cos ϕ ( r, θ ) cos r ) p − cos r cos ϕ ( r, θ ) . Proof . Let u = cos r cos ϕ , ∂∂ϕ ϕ − cos r sin ϕ arccos(cos ϕ cos r ) p − cos r cos ϕ ! = sin r − u ( r, θ ) − u ( r, θ ) arccos u ( r, θ ) p − u ( r, θ ) ! is positive, thus θ = ϕ − cos r sin ϕ arccos(cos ϕ cos r ) √ − cos r cos ϕ is a bijection from [ − π, π ] onto itself,hence the uniqueness. (cid:3) ϕ ( r, θ ), f ′′ ( iϕ ( r, θ )) = 2 sin r − u ( r, θ ) − u ( r, θ ) arccos u ( r, θ ) p − u ( r, θ ) ! , is positive, where u ( r, θ ) = cos r cos ϕ ( r, θ ).By using the steepest descent method we can deduce Proposition 3.10
Let r ∈ (0 , π ) , θ ∈ [ − π, π ] . Then when t → , p t ( r, θ ) ∼ t → πt ) n + sin r (arccos u ( r, θ )) n r − u ( r,θ ) arccos u ( r,θ ) √ − u ( r,θ ) e − ( ϕ ( r,θ )+ θ )2 tan2 r t sin2( ϕ ( r,θ )) (1 − u ( r, θ ) ) n − , where u ( r, θ ) = cos r cos ϕ ( r, θ ) . Remark 3.11
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