The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration
In this note, we study the sub-Laplacian of the 15-dimensional octonionic anti-de Sitter space, which is obtained by lifting with respect to the anti-de Sitter fibration the Laplacian of the octonionic hyperbolic space ¹. We also obtain two integral representations for the corresponding subelliptic...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2021 |
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Інститут математики НАН України
2021
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| Цитувати: | The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration. Fabrice Baudoin and Gunhee Cho. SIGMA 17 (2021), 014, 9 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860263583088115712 |
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| author | Baudoin, Fabrice Cho, Gunhee |
| author_facet | Baudoin, Fabrice Cho, Gunhee |
| citation_txt | The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration. Fabrice Baudoin and Gunhee Cho. SIGMA 17 (2021), 014, 9 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this note, we study the sub-Laplacian of the 15-dimensional octonionic anti-de Sitter space, which is obtained by lifting with respect to the anti-de Sitter fibration the Laplacian of the octonionic hyperbolic space ¹. We also obtain two integral representations for the corresponding subelliptic heat kernel.
|
| first_indexed | 2026-03-21T09:33:20Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 014, 9 pages
The Subelliptic Heat Kernel
of the Octonionic Anti-De Sitter Fibration
Fabrice BAUDOIN and Gunhee CHO
Department of Mathematics, University of Connecticut,
196 Auditorium Road, Storrs, CT 06269-3009, USA
E-mail: fabrice.baudoin@uconn.edu, gunhee.cho@uconn.edu
Received July 31, 2020, in final form January 29, 2021; Published online February 10, 2021
https://doi.org/10.3842/SIGMA.2021.014
Abstract. In this note, we study the sub-Laplacian of the 15-dimensional octonionic anti-de
Sitter space which is obtained by lifting with respect to the anti-de Sitter fibration the Lapla-
cian of the octonionic hyperbolic space OH1. We also obtain two integral representations
for the corresponding subelliptic heat kernel.
Key words: sub-Laplacian; 15-dimensional octonionic anti-de Sitter space; the anti-de Sitter
fibration
2020 Mathematics Subject Classification: 58J35; 53C17
1 Introduction and results
In this note we study the sub-Laplacian and the corresponding sub-Riemannian heat kernel of
the octonionic anti-de Sitter fibration
S7 ↪→ AdS15(O)→ OH1.
This paper follows the previous works [2, 3, 10] which respectively concerned:
1. The complex anti-de Sitter fibrations:
S1 ↪→ AdS2n+1(C)→ CHn.
2. The quaternionic anti-de Sitter fibrations:
S3 ↪→ AdS4n+3(H)→ HHn.
The 15-dimensional anti-de Sitter fibration is the last model space that remained to be
studied of a sub-Riemannian manifold arising from a H-type semi-Riemannian submersion over
a rank-one symmetric space, see the Table 3 in [4].
Similarly to the complex and quaternionic case, the sub-Laplacian is defined as the lift on
AdS15(O) of the Laplace–Beltrami operator of the octonionic hyperbolic space OH1. However,
in the complex and quaternionic case the Lie group structure of the fiber played an important
role that we can not use here, since the fiber S7 is not a group. Instead, we make use of some
algebraic properties of S7 that were already pointed out and used by the authors in [1] for the
study of the octonionic Hopf fibration:
S7 ↪→ S15 → OP 1.
Let us briefly describe our main results. Due to the cylindrical symmetries of the fibration,
the heat kernel of the sub-Laplacian only depends on two variables: the variable r which is the
mailto:fabrice.baudoin@uconn.edu
mailto:gunhee.cho@uconn.edu
https://doi.org/10.3842/SIGMA.2021.014
2 F. Baudoin and G. Cho
Riemannian distance on OH1 (the starting point is specified with inhomogeneous coordinate in
Section 3) and the variable η which is the Riemannian distance starting at a pole on the fiber S7.
We prove in Proposition 3.1 that in these coordinates, the radial part of the sub-Laplacian L̃
writes
L̃ =
∂2
∂r2
+ (7 coth r + 7 tanh r)
∂
∂r
+ tanh2 r
(
∂2
∂η2
+ 6 cot η
∂
∂η
)
.
As a consequence of this expression for the sub-Laplacian, we are able to derive two equivalent
formulas for the heat kernel. The first formula, see Proposition 4.1, reads as follows: for r ≥ 0,
η ∈ [0, π), t > 0
pt(r, η) =
∫ ∞
0
st(η, iu)qt,15(cosh r coshu) sinh6 udu,
where st is the heat kernel of the Jacobi operator
4̃S7 =
∂2
∂η2
+ 6 cot η
∂
∂η
with respect to the measure sin6 η dη, and where qt,15 is the Riemannian heat kernel on the 15-
dimensional real hyperbolic space H15 given in (4.1). The second formula, see Proposition 4.2,
writes as follows:
pt(r, η) =
∫ π
0
∫ ∞
0
Gt(η, ϕ, u)qt,9(cosh r coshu) sin5 ϕdudϕ,
where qt,9 is Riemannian heat kernel on the 9-dimensional hyperbolic space H9 and Gt(η, ϕ, u)
is given in (4.3).
Similarly to [2, 3, 10], it might be expected that explicit integral representations of the
heat kernel might be used to study small-time asymptotics, inside and outside of the cut-locus.
Integral representations of heat kernels can also be used to obtain sharp heat kernel estimates,
see [7]. Those applications of the heat kernel representations we obtain will possibly be addressed
in a future research project.
2 The octonionic anti-de Sitter fibration
Let
O =
x =
7∑
j=0
xjej , xj ∈ R
,
be the division algebra of octonions (see [9] for explicit representations of this algebra). We
recall that the multiplication rules are given by
eiej = ej if i = 0,
eiej = ei if j = 0,
eiej = −δije0 + εijkek otherwise,
where δij is the Kronecker delta and εijk is the completely antisymmetric tensor with value 1
when ijk = 123, 145, 176, 246, 257, 347, 365 (also see [1]). The octonionic norm is defined for
x ∈ O by
||x||2 =
7∑
j=0
x2j .
The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration 3
The octonionic anti-de Sitter space AdS15(O) is the quadric defined as the pseudo-hyperbolic
space by:
AdS15(O) =
{
(x, y) ∈ O2, ||(x, y)||2O = −1
}
,
where
||(x, y)||2O := ||x||2 − ||y||2.
In real coordinates we have x =
∑7
j=0 xjej , y =
∑7
j=0 yjej , and the pseudo-norm can be
written as
x20 + · · ·+ x27 − y20 − · · · − y27.
As such, AdS15(O) is embedded in the flat 16-dimensional space R8,8 endowed with the Lo-
rentzian real signature (8, 8) metric
ds2 = dx20 + · · ·+ dx27 − dy20 − · · · − dy27.
Consequently, AdS15(O) is naturally endowed with a pseudo-Riemannian structure of signa-
ture (8, 7).
Let OH1 denote the octonionic hyperbolic space. The map π : AdS15(O) → OH1, given by
(x, y) 7→ [x : y] = y−1x is a pseudo-Riemannian submersion with totally geodesic fibers isometric
to the seven-dimensional sphere S7. Notice that, as a topological manifold, OH1 can therefore
be identified with the unit open ball in O. The pseudo-Riemannian submersion π yields the
octonionic anti-de Sitter fibration
S7 ↪→ AdS15(O)→ OH1.
For further information on semi-Riemannian submersions over rank-one symmetric spaces, we
refer to [6].
3 Cylindrical coordinates and radial part of the sub-Laplacian
The sub-Laplacian L on AdS15(O) we are interested in is the horizontal Laplacian of the Rieman-
nian submersion π : AdS15(O)→ OH1, i.e., the horizontal lift of the Laplace–Beltrami operator
of OH1. It can be written as
L = �AdS15(O) +4V , (3.1)
where �AdS15(O) is the d’Alembertian, i.e., the Laplace–Beltrami operator of the pseudo-Rie-
mannian metric and 4V is the vertical Laplacian. Since the fibers of π are totally geodesic and
isometric to S7 ⊂ AdS15(O), we note that �AdS15(O) and 4V are commuting operators, and we
can identify
4V = 4S7 . (3.2)
The sub-Laplacian L is associated with a canonical sub-Riemannian structure on AdS15(O)
which is of H-type, see [4].
To study L, we introduce a set of coordinates that reflect the cylindrical symmetries of the
octonionic unit sphere which provides an explicit local trivialization of the octonionic anti-de
Sitter fibration. Consider the coordinates w ∈ OH1, where w is the inhomogeneous coordinate
on OH1 given by w = y−1x, with x, y ∈ AdS15(O). Consider the north pole p ∈ S7 and take
4 F. Baudoin and G. Cho
Y1, . . . , Y7 to be an orthonormal frame of TpS7. Let us denote expp the Riemannian exponential
map at p on S7. Then the cylindrical coordinates we work with are given by
(w, θ1, . . . , θ7) 7→
(
expp
(∑7
i=1 θiYi
)
w√
1− ρ2
,
expp
(∑7
i=1 θiYi
)√
1− ρ2
)
∈ AdS15(O),
where ρ = ‖w‖ and ‖θ‖ =
√
θ21 + · · ·+ θ27 < π.
A function f on AdS15(O) is called radial cylindrical if it only depends on the two coordinates
(ρ, η) ∈ [0, 1)× [0, π] where η =
√∑7
i=1 θ
2
i . More precisely f is radial cylindrical if there exists
a function g so that
f
(
expp
(∑7
i=1 θiYi
)
w√
1− ρ2
,
expp
(∑7
i=1 θiYi
)√
1− ρ2
)
= g(ρ, η).
We denote by D the space of smooth and compactly supported functions on [0, 1) × [0, π).
Then the radial part of L is defined as the operator L̃ such that for any f ∈ D, we have
L(f ◦ ψ) =
(
L̃f
)
◦ ψ. (3.3)
We now compute L̃ in cylindrical coordinates.
Proposition 3.1. The radial part of the sub-Laplacian on AdS15(O) is given in the coordinates
(r, η) by the operator
L̃ =
∂2
∂r2
+ (7 coth r + 7 tanh r)
∂
∂r
+ tanh2 r
(
∂2
∂η2
+ 6 cot η
∂
∂η
)
,
where r = tanh−1 ρ is the Riemannian distance on OH1 from the origin.
Proof. Note that the radial part of the Laplace–Beltrami operator on the octonionic hyperbolic
space OH1 is
4̃OH1 =
∂2
∂r2
+ (7 coth r + 7 tanh r)
∂
∂r
,
and the radial part of the Laplace–Beltrami operator on S7 is
4̃S7 =
∂2
∂η2
+ 6 cot η
∂
∂η
. (3.4)
Since the octonionic anti-de Sitter fibration defines a totally geodesic submersion with base
space OH1 and fiber S7, the semi-Riemannian metric on AdS15(O) is locally given by a warped
product between the Riemannian metric of OH1 and the Riemannian metric on S7. Hence the
radial part of the d’Alembertian becomes
�̃AdS15(O) =
∂2
∂r2
+ (7 coth r + 7 tanh r)
∂
∂r
+ g(r)
(
∂2
∂η2
+ 6 cot η
∂
∂η
)
, (3.5)
for some smooth function g to be computed.
On the other hand, from the isometric embedding AdS15(O) ⊂ O×O, the d’Alembertian on
AdS15(O) is a restriction of the d’Alembertian on O×O ' R8,8 in the sense that for a smooth
f : AdS15(O)→ R
�AdS15(O)f = �O×Of
∗
/AdS15(O)
,
The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration 5
where �O×O =
∑7
i=0
(
∂2
∂x2i
− ∂2
∂y2i
)
and for x, y ∈ O such that ‖y‖2 − ‖x‖2 > 0, f∗(x, y) =
f
(
x√
‖y‖2−‖x‖2
, y√
‖y‖2−‖x‖2
)
. For the specific choice of the function f(x, y) = y1, one easily
computes that �O×Of
∗
/AdS15(O)
(x, y) = 15y1, thus
�AdS15(O)f(x, y) = 15y1.
For the point with coordinates(
expp
(∑7
i=1 θiYi
)
w√
1− ρ2
,
expp
(∑7
i=1 θiYi
)√
1− ρ2
)
∈ AdS15(O)
one has
y1 =
cos η√
1− ρ2
= cosh r cos η.
We therefore deduce that
�̃AdS15(O)(cosh r cos η) = 15 cosh r cos η.
Using the formula (3.5), after a straightforward computation, this yields g(r) = − 1
cosh2 r
and
therefore
�̃AdS15(O) =
∂2
∂r2
+ (7 coth r + 7 tanh r)
∂
∂r
− 1
cosh2 r
(
∂2
∂η2
+ 6 cot η
∂
∂η
)
= 4̃OH1 −
1
cosh2 r
4̃S7 .
Finally, to conclude, one notes that the sub-Laplacian L is given by the difference between the
Laplace–Beltrami operator of AdS15(O) and the vertical Laplacian. Therefore by (3.1) and (3.2),
L̃ = �̃AdS15(O) + 4̃S7 =
∂2
∂r2
+ (7 coth r + 7 tanh r)
∂
∂r
+ tanh2 r
(
∂2
∂η2
+ 6 cot η
∂
∂η
)
. �
Remark 3.2. As a consequence of the previous result, we can check that the Riemannian
measure of AdS15(O) in the coordinates (r, η), which is the symmetric and invariant measure
for L̃ is given by
dµ =
π7
90
sinh7 r cosh7 r sin6 η dr dη. (3.6)
(See also Remark 2 in [1], which corresponds to the case of the octonionic Hopf fibration.)
4 Integral representations of the subelliptic heat kernel
In this section, we give two integral representations of the subelliptic heat kernel associated
with L̃. We denote by pt(r, η) the heat kernel of L̃ issued from the point r = η = 0 with respect
to the measure (3.6). We remark that studying the subelliptic heat kernel associated with L̃ is
enough to study the heat kernel of L, because due to (3.3) the heat kernel ht(w, θ) of L issued
from the point with cylindric coordinates w = 0, θ = 0 is then given by
ht(w, θ) = pt
(
tanh−1 ‖w‖, ‖θ‖
)
.
6 F. Baudoin and G. Cho
4.1 First integral representation
We denote by st the heat kernel of the operator
4̃S7 =
∂2
∂η2
+ 6 cot η
∂
∂η
with respect to the reference measure sin6 η dη. The operator 4̃S7 belongs to the family of
Jacobi diffusion operators which have been extensively studied in the literature, see for instance
the appendix in [5] and the references therein. In particular, the spectrum of 4̃S7 is given by
Sp
(
−4̃S7
)
= {m(m+ 6), m ∈ N},
and the eigenfunction corresponding to the eigenvalue m(m+6) is P
5/2,5/2
m (cos η) where P
5/2,5/2
m
is the Jacobi polynomial
P 5/2,5/2
m (x) =
(−1)m
2mm!
(
1− x2
)5/2 dm
dxm
(
1− x2
)5/2+m
.
As a consequence, one has the following spectral decomposition for the heat kernel:
st(η, u) =
1
π
+∞∑
m=0
24m+7m!(m+ 5)![(m+ 3)!]2
(2m+ 6)!(2m+ 5)!
e−m(m+6)tP 5/2,5/2
m (cos η)P 5/2,5/2
m (cosu).
Proposition 4.1. For r ≥ 0, η ∈ [0, π], and t > 0 we have
pt(r, η) =
∫ ∞
0
st(η, iu)qt,15(cosh r coshu) sinh6 udu,
where
qt,15(cosh s) :=
e−49t
(2π)7
√
4πt
(
− 1
sinh s
d
ds
)7
e−s
2/4t (4.1)
is the Riemannian heat kernel on the 15-dimensional real hyperbolic space H15.
Proof. Since π : AdS15(O) → OH1 is a (semi-Riemannian) totally geodesic submersion, the
operators �̃AdS15(O) and 4̃S7 commute. Thus
etL̃ = e
t(�̃AdS15(O)+4̃S7 ) = et4̃S7 e
t�̃AdS15(O) .
We deduce that the heat kernel of L̃ can be written as
pt(r, η) =
∫ π
0
st(η, u)p
�̃AdS15(O)
t (r, u) sin6 u du, (4.2)
where st is the heat kernel of (3.4) with respect to the measure sin6 η dη, η ∈ [0, π), and
p
�̃AdS15(O)
t (r, u) the heat kernel at (0, 0) of �̃AdS15(O) with respect to the measure in (3.6), i.e.,
dµ(r, u) =
π7
90
sinh7 r cosh7 r sin6 udr du, r ∈ [0,∞), u ∈ [0, π].
In order to write (4.2) more precisely, let us consider the analytic change of variables τ : (r, η)→
(r, iη) that will be applied on functions of the type f(r, η) = h(r)e−iλη, with h smooth and
The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration 7
compactly supported on [0,∞) and λ > 0. Then as we saw in the proof of Proposition 3.1 one
can see that
�̃AdS15(O)(f ◦ τ) =
(
4̃H15f
)
◦ τ,
where
4̃H15 = 4̃OH1 +
1
cosh2 r
4̃P , 4̃P =
∂2
∂η2
+ 6 coth η
∂
∂η
.
Then, one deduces
etL̃(f ◦ τ) = et4̃S7 e
t�̃AdS15(O)(f ◦ τ) = et4̃S7
((
et4̃H15f
)
◦ τ
)
=
(
e−t4̃P et4̃H15f
)
◦ τ.
Now, since for every f(r, η) = h(r)e−iλη,(
e
t�̃AdS15(O)f
)
(0, 0) =
(
et4̃H15
)(
f ◦ τ−1
)
(0, 0),
one deduces that for a function h depending only on u,∫ π
0
h(u)p
�̃AdS15(O)
t (r, u) sin6 u du =
∫ ∞
0
h(−iu)qt,15(cosh r coshu) sinh6 udu.
Therefore, coming back to (4.2), one infers that using the analytic extension of st one must have∫ π
0
st(η, u)p
�AdS15(O)
t (r, u) sin6 u du =
∫ ∞
0
st(η,−iu)qt,15(cosh r coshu) sinh6 udu,
where qt,15 is the Riemannian heat kernel on the real hyperbolic space H15 given in (4.1). �
4.2 Second integral representation
Proposition 4.2. For r ≥ 0, η ∈ [0, π], and t > 0 we have
pt(r, η) =
∫ π
0
∫ ∞
0
Gt(η, ϕ, u)qt,9(cosh r coshu) sin5 ϕdu dϕ.
where qt,9 is the 9-dimensional Riemannian heat kernel on the hyperbolic space H9:
qt,9(cosh s) :=
e−16t
(2π)4
√
4πt
(
1
sinh s
d
ds
)4
e−s
2/4t,
and
Gt(η, ϕ, u) =
15
8
∑
m≥0
e−(m(m+6)+33)t(cos η + i sin η cosϕ)m cosh((m+ 3)u). (4.3)
Proof. The strategy of the following method appeals to some results proved in [8]. Firstly, we
decompose the subelliptic heat kernel in the η variable with respect to the basis of normalized
eigenfunctions of 4̃S7 = ∂2
∂η2
+ 6 cot η ∂
∂η . Accordingly,
pt(r, η) =
∑
m≥0
fm(t, r)hm(η),
where for each m, hm is given by
hm(η) =
15
16
∫ π
0
(cos η + i sin η cosϕ)msin5 ϕdϕ
8 F. Baudoin and G. Cho
and fm(t, ·) solves the following heat equation
∂
∂t
fm(t, r) =
(
∂2
∂r2
+ (7 coth r + 7 tanh r)
∂
∂r
−m(m+ 6)tanh2 r
)
fm(t, r)
=
(
∂2
∂r2
+ (7 coth r + 7 tanh r)
∂
∂r
+
m(m+ 6)
cosh2 r
−m(m+ 6)
)
fm(t, r).
We consider then the operator
Lm :=
∂2
∂r2
+ (7 coth r + 7 tanh r)
∂
∂r
+
m(m+ 6)
cosh2 r
+ 49,
which was studied in [8, p. 229]. From [8, Theorem 2], with α = 3 + m
2 , β = −m
2 , we deduce
that the solution to the wave Cauchy problem associated with the subelliptic Laplacian is given
f ∈ C∞0
(
OH1
)
by
cos
(
s
√
−Lm
)
(f)(w) =
− sinh s
(2π)4
(
1
sinh s
d
ds
)4 ∫
OH1
Km(s, w, y)f(y)
dy(
1− ||y||2
)8 ,
where
Km(s, w, y) =
(1− 〈w, y〉)3+m/2
(1− 〈w, y〉)m/2
1
cosh3(d(w, y))
√
cosh2(s)− cosh2(d(w, y))
× 2F1
(
m+ 3,−m− 3,
1
2
;
cosh(d(w, y))− cosh(s)
2 cosh(d(w, y))
)
,
where 2F1 is the Gauss hypergeometric function and dy stands for the Lebesgue measure in R8.
Using the spectral formula
etL =
1√
4πt
∫
R
e−s
2/(4t) cos
(
s
√
−L
)
ds,
which holds for any non positive self-adjoint operator, we deduce that the solution to the heat
Cauchy problem associated with Lm:
etLm(f)(w) =
e−m(m+6)t−72t
√
4πt(2π)4
∫
R
ds(− sinh s)e−s
2/(4t)
×
(
1
sinh s
d
ds
)4 ∫
OH1
Km(s, w, y)f(y)
dy(
1− ||y||2
)8 .
Performing integration by parts 4-times,∫
R
ds(− sinh s)
(
1
sinh s
d
ds
)4
e−s
2/(4t)
∫
OH1
Km(s, w, y)f(y)
dy(
1− ||y||2
)8
=
∫
OH1
f(y)
dy
(1− ||y||2)8
∫
R
ds(− sinh s)Km(s, w, y)
(
1
sinh s
d
ds
)4
e−s
2/4t
= 2
∫
OH1
f(y)
dy(
1− ||y||2
)8 ∫ ∞
d(w,y)
d(cosh(s))Km(s, w, y)
(
1
sinh s
d
ds
)4
e−s
2/4t.
Thus we get
etLm(f)(0) = 2e−(m(m+6)+33)t
∫
OH1
f(y)
dy(
1− ||y||2
)8 ∫ ∞
d(0,y)
d(cosh s)Km(s, 0, y)qt,9(cosh s).
The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration 9
As a result, the subelliptic heat kernel of Lm reads
dy(
1− ||y||2
)8 ∫ ∞
d(0,y)
d(cosh s)Km(s, 0, y)qt,9(cosh s)
= dr sinh7 r cosh7 r
∫ ∞
r
d(cosh s)Km(s, 0, y)qt,9(cosh s).
By changing the variable cosh s = cosh r coshu for u ≥ 0, the last expression becomes
dr sinh7 r cosh7 r
∫ ∞
0
2F1
(
m+ 3,−m− 3,
1
2
;
1− coshu
2
)
qt,9(cosh r coshu) du.
Therefore pt(r, η) has the integral representation
2
∑
m≥0
e−(m(m+6)+33)thm(η)
∫ ∞
0
2F1
(
m+ 3,−m− 3,
1
2
;
1− coshu
2
)
qt,9(cosh r coshu) du.
Now, notice that 2F1
(
m+ 3,−m− 3, 12 ; 1−coshu
2
)
is simply the Cheybyshev polynomial of the
first kind
Tm+3(x) = 2F1
(
m+ 3,−m− 3,
1
2
;
1− x
2
)
,
for all x ∈ C. Therefore, one has
2F1
(
m+ 3,−m− 3,
1
2
;
1− coshu
2
)
= Tm+3(coshu) = cosh((m+ 3)u),
and the proof is over. �
Acknowledgements
F.B. is partially funded by the NSF grant DMS-1901315.
References
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https://doi.org/10.1007/s11118-020-09854-4
https://arxiv.org/abs/1904.08568
https://doi.org/10.1007/s00013-018-1201-1
https://arxiv.org/abs/1802.04199
https://doi.org/10.1007/s11118-018-9746-y
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https://doi.org/10.1023/A:1006501627929
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https://arxiv.org/abs/1204.3642
1 Introduction and results
2 The octonionic anti-de Sitter fibration
3 Cylindrical coordinates and radial part of the sub-Laplacian
4 Integral representations of the subelliptic heat kernel
4.1 First integral representation
4.2 Second integral representation
References
|
| id | nasplib_isofts_kiev_ua-123456789-211174 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T09:33:20Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Baudoin, Fabrice Cho, Gunhee 2025-12-25T13:22:15Z 2021 The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration. Fabrice Baudoin and Gunhee Cho. SIGMA 17 (2021), 014, 9 pages 1815-0659 2020 Mathematics Subject Classification: 58J35; 53C17 arXiv:2003.13512 https://nasplib.isofts.kiev.ua/handle/123456789/211174 https://doi.org/10.3842/SIGMA.2021.014 In this note, we study the sub-Laplacian of the 15-dimensional octonionic anti-de Sitter space, which is obtained by lifting with respect to the anti-de Sitter fibration the Laplacian of the octonionic hyperbolic space ¹. We also obtain two integral representations for the corresponding subelliptic heat kernel. F.B. is partially funded by the NSF grant DMS-1901315. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration Article published earlier |
| spellingShingle | The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration Baudoin, Fabrice Cho, Gunhee |
| title | The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration |
| title_full | The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration |
| title_fullStr | The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration |
| title_full_unstemmed | The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration |
| title_short | The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration |
| title_sort | subelliptic heat kernel of the octonionic anti-de sitter fibration |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211174 |
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