Post on 29-Jun-2020
Definition and basic properties of heat
kernels III, Constructions
Zhiqin Lu,Department of Mathematics,UC Irvine, Irvine CA 92697
April 27, 2010
TheoremLet M be a complete Riemannian manifold. Then there is aheat kernel
H(x, y, t) ∈ C∞(M ×M × R+),
such that
(e∆tf)(x) =
∫M
H(x, y, t)f(y)dy
satisfying
1 H(x, y, t) = H(y, x, t),
2 limt→0+
H(x, y, t) = δx(y);
3 (∆− ∂∂t
)H = 0;
4 H(x, y, t) =∫MH(x, z, t− s)H(z, y, s)dz for any
0 < s ≤ t.
The proof composed of two steps
1 Construct the paramatrix;
2 Using the paramatrix to construct the heat kernel.
We only formally construct the heat kernel.
The k-simplex ∆k is the following subset of Rk
{(t1, · · · , tk) | 0 ≤ t1 ≤ · · · ≤ tk ≤ 1}.
For t > 0, we write t∆k for the rescaled simplex
{(t1, · · · , tk) | 0 ≤ t1 ≤ · · · ≤ tk ≤ t}.
We only formally construct the heat kernel.
The k-simplex ∆k is the following subset of Rk
{(t1, · · · , tk) | 0 ≤ t1 ≤ · · · ≤ tk ≤ 1}.
For t > 0, we write t∆k for the rescaled simplex
{(t1, · · · , tk) | 0 ≤ t1 ≤ · · · ≤ tk ≤ t}.
We assume that U(x, y, t) be a function on M ×M ×R+ suchthat limt→0+ U(x, y, t) = δx(y) be the the Dirac function. Let
R(x, y, t) =dU(x, y, t)
dt−∆xU(x, y, t),
where ∆x means the Laplace operator for the variable x. Notethat formally, any function g(x, y, t) defines one-parameterfamily of operators Gt by the formula
Gtf(x) =
∫M
g(x, y, t)f(y)dy.
We use the Rt, Ut to denote the corresponding families ofoperators with respect to the functions R(x, y, t) andU(x, y, t), respectively. For any k ≥ 1, define the operator
Qkt =
∫t∆k
Ut−tkRtk−tk−1· · ·Rt2−t1Rt1dt1 · · · dtk,
and Q0t = Ut. Let
R(k)(s) =
∫s∆k−1
Rs−tk−1· · ·Rt2−t1Rt1dt1 · · · dtk−1,
and R(0)(s) = 0.
Since the derivative of the integral of the form∫ t0a(t− s)b(s)ds is equal to∫ t
0
da
dt(t− s)b(s)ds+ a(0)b(t),
we have (∂
∂t−∆
)Qkt = R(k+1)(t) +R(k)(t).
As a result, we have(∂
∂t−∆
) ∞∑k=0
(−1)kQkt = 0.
We wish to find the the following form of the fundamentalsolution of the heat equation:
U(x, y, t) ∼ (4πt)−n2 e−d
2(x,y)/4t
{∑i≥0
φi(x, y)ti
}
where d(x, y) is the distance function on the Riemannianmanifold. U(x, y, t) should satisfy
1 limt→0+ U(x, y, t) = δx(y), where δx(y) is the Diracfunction at x;
2 For any N , limt→0+ (∆− ∂∂t
)U(x, y, t) = O(tN).
The function U(x, y, t) is called the paramatrix of the heatkernel.
We pick a normal coordinate system (y1, · · · , yn), and letr = d(x, y) be the Riemannian distance. We identify aneighborhood of y to a small ball of Ty(M) by the exponentialmap. Under this map, the coordinates of x can be written as(x1, · · · , xn). On the other hand, let (θ1, · · · , θn−1) be acoordinate system on Sn−1, then (r, θ1, · · · , θn−1) gives acoordinate system at y also, and this coordinate system iscalled the polar coordinates.
Let ψ(r) be a function of r, and let g = det(sij). Then wehave
∆ψ =d2ψ
dr2+
(d log
√g
dr
)dψ
dr,
∆(φψ) = φ∆ψ + ψ∆φ+ 2dφ
dr
dψ
dr.
We let
ψ =1
(4πt)n/2e−
r2
4t ,
φ = φ0 + φ1t+ · · ·+ φN tN ,
where φj = φj(x, y) are smooth functions on M ×M , and
uN = ψφ =1
(4πt)n/2e−
r2
4t
N∑i=0
φiti.
Then(∆− ∂
∂t
)uN = φ
(∆ψ − ∂ψ
∂t
)+ ψ
(∆φ− ∂φ
∂t
)+ 2
∂φ
∂r
dψ
dr
Since
∆ψ − ∂ψ
∂t=d log
√g
dr
dψ
dr,
dψ
dr= − r
2tψ.
we have(∆− ∂
∂t
)uN =
ψ
t
N∑k=0
[∆φk−1 −
(k +
r
2
d log√g
dr
)φk − r
dφkdr
]tk.
Thus in order to find the paramatrix, we set
rdφkdr
+
(k +
r
2
d log√g
dr
)φk = ∆φk−1
for k = 0, · · · , N , where we let φ−1 = 0. The solutions of theabove ODEs are
φ0(x, y) = g−14 (x);
φk(x, y) = g−14 (x)r(x, y)−k
×∫ r(x,y)
0
rk−1(∆φk−1)
(rx
r(x, y)
)g
(rx
r(x, y)
) 14
dr.
Thus we have (∆− ∂
∂t
)uN =
ψ
t(∆φN)tN .
From the above, we proved that
LemmaThere is a unique formal solution U(x, y, t) of the heatequation (
∆− ∂
∂t
)U(x, y, t) = 0
of the form
uN = ψφ =1
(4πt)n/2e−
r2
4t
N∑i=0
φiti.
Let η be a smooth function such that η = 1 for t < 1 andη = 0 for t > 2. Let
p(x, y) = η
(2r(x, y)
δ
)be the cut-off function, where δ is the injectivity radius. Thenfor any N , we consider the functionuN(x, y, t) = p(x, y)uN(x, y, t). We shall prove that
LemmaFor any N sufficiently large, we have
1 limt→0+ uN(x, y, t) = δx(y);
2 The kernel RN(x, y, t) =(∆y − ∂
∂t
)uN(x, y, t) satisfies
the estimate
||RN(x, y, t)||Cl ≤ C(l)tN−l/2−1.
Using the above estimate, we know that for N � 0, we have
||RN(x, y, t)||Cl ≤ Ctα.
It follows that
||R(k)(s)||Cl ≤ Ckα!
(α + k)!.
Since ∑k
Ckα!
(α + k)!< +∞,
our formal construction is convergent to the heat kernel.
TheoremThe ∆p on Lp(M) is well defined as the infinitesimalgenerator of the heat semi-group.
A short-cut
The following equation is true
H(x, y, t) = U(x, y, t)−∫ t
0
e∆x(t−s)(∂
∂s−∆x
)U(x, y, s)ds
Two problems
1 The self-adjoint extension of ∆ has to be assumed first;
2 The convergence of U(x, y, t).
A short-cut
The following equation is true
H(x, y, t) = U(x, y, t)−∫ t
0
e∆x(t−s)(∂
∂s−∆x
)U(x, y, s)ds
Two problems
1 The self-adjoint extension of ∆ has to be assumed first;
2 The convergence of U(x, y, t).
Let u be a positive harmonic function on M .
Defineφ = |∇ log u|.Then we have
φi =1
u|∇u|ujuji −
|∇u|u2
ui
∆φ = φii =1
u|∇u|∑
u2ji +
1
u|∇u|ujujii
− 2
u2|∇u|uiujuij −
1
u|∇u|3∑i
(∑j
ujuji
)2
+ 2φ3
Let u be a positive harmonic function on M .Defineφ = |∇ log u|.
Then we have
φi =1
u|∇u|ujuji −
|∇u|u2
ui
∆φ = φii =1
u|∇u|∑
u2ji +
1
u|∇u|ujujii
− 2
u2|∇u|uiujuij −
1
u|∇u|3∑i
(∑j
ujuji
)2
+ 2φ3
Let u be a positive harmonic function on M .Defineφ = |∇ log u|.Then we have
φi =1
u|∇u|ujuji −
|∇u|u2
ui
∆φ = φii =1
u|∇u|∑
u2ji +
1
u|∇u|ujujii
− 2
u2|∇u|uiujuij −
1
u|∇u|3∑i
(∑j
ujuji
)2
+ 2φ3
We assume at a point uj = 0 for j > 1. Then we have
φ1 =1
u|∇u|u1u11 −
|∇u|u2
u1
∆φ =1
u|∇u|∑
u2ji +
1
u|∇u|ujujii
− 2u1
u2u11 −
1
u|∇u|∑i
u21i + 2φ3
We have
1
u|∇u|ujujii ≥
1
u|∇u|uj(∆u)j −Kφ
and
1
u|∇u|∑
u2ji −
1
u|∇u|∑i
u21i
≥ 1
u|∇u|∑j≥2
u2jj ≥
1
(n− 1)u|∇u|u2
11
We have
1
u|∇u|ujujii ≥
1
u|∇u|uj(∆u)j −Kφ
and
1
u|∇u|∑
u2ji −
1
u|∇u|∑i
u21i
≥ 1
u|∇u|∑j≥2
u2jj ≥
1
(n− 1)u|∇u|u2
11
Thus we have
∆φ ≥ 2φ3 − 2u1
u2u11 +
1
(n− 1)u|∇u|u2
11 −Kφ
We have
φ3 − |∇u|u2
u11 = −∇φ∇uu
Thus we have
∆φ ≥ A∇φ∇uu
+ (2 + A)φ3 − (2 + A)u1
u2u11 +
1
(n− 1)u|∇u|u2
11 −Kφ
If we choose A close to 2 enough, we can find an ε > 0 suchthat
∆φ ≥ A∇φ∇uu
+1
2εφ3 −Kφ.
Thus we have
∆φ ≥ 2φ3 − 2u1
u2u11 +
1
(n− 1)u|∇u|u2
11 −Kφ
We have
φ3 − |∇u|u2
u11 = −∇φ∇uu
Thus we have
∆φ ≥ A∇φ∇uu
+ (2 + A)φ3 − (2 + A)u1
u2u11 +
1
(n− 1)u|∇u|u2
11 −Kφ
If we choose A close to 2 enough, we can find an ε > 0 suchthat
∆φ ≥ A∇φ∇uu
+1
2εφ3 −Kφ.
Thus we have
∆φ ≥ 2φ3 − 2u1
u2u11 +
1
(n− 1)u|∇u|u2
11 −Kφ
We have
φ3 − |∇u|u2
u11 = −∇φ∇uu
Thus we have
∆φ ≥ A∇φ∇uu
+ (2 + A)φ3 − (2 + A)u1
u2u11 +
1
(n− 1)u|∇u|u2
11 −Kφ
If we choose A close to 2 enough, we can find an ε > 0 suchthat
∆φ ≥ A∇φ∇uu
+1
2εφ3 −Kφ.
The Bakry-Emery geometry was introduced to study thediffusion processes. For a Riemannian manifold (M, g) and asmooth function φ on M , the Bakry-Emery manifold is a triple(M, g, φ), where the measure on M is the weighted measuree−φdVg. The Bakry-Emery Ricci curvature is defined to be
Ric∞ = Ric + Hess(φ),
and the Bakry-Emery Laplacian is
∆φ = ∆−∇φ · ∇.
The operator can be extended as a self-adjoint operator withrespect to the weighted measure e−φdVg.
The B-E Laplacian
We have ∫∆φfge
−φ = −∫∇f∇g · e−φ
Therefore, it can be extended as a self-adjoint ellipticoperator. Hodge theorem is also valid for this new operator.
The B-E Laplacian
We have ∫∆φfge
−φ = −∫∇f∇g · e−φ
Therefore, it can be extended as a self-adjoint ellipticoperator.
Hodge theorem is also valid for this new operator.
The B-E Laplacian
We have ∫∆φfge
−φ = −∫∇f∇g · e−φ
Therefore, it can be extended as a self-adjoint ellipticoperator. Hodge theorem is also valid for this new operator.
Theorem (Brighton)Let f be a bounded function and let u be a f -harmonicfunction.
That is∆u−∇f∇u = 0
Then if M is a Ricci nonnegative manifold and u is positive, umust be a constant.
Theorem (Brighton)Let f be a bounded function and let u be a f -harmonicfunction.That is
∆u−∇f∇u = 0
Then if M is a Ricci nonnegative manifold and u is positive, umust be a constant.
Define a function U(x, y) on M × R+.
U(x, y) = u(x)− 1
2y2∇f∇u
Then∆U = 0
at (x, 0).Let Φ = |∇ logU |. Then we have
∆Φ ≥ AΦ3 + C∇Φ∇UU
for A > 0 and ∆ the Laplacian on the n+ 1 dimensionalmanifold M × R+.
Define a function U(x, y) on M × R+.
U(x, y) = u(x)− 1
2y2∇f∇u
Then∆U = 0
at (x, 0).Let Φ = |∇ logU |. Then we have
∆Φ ≥ AΦ3 + C∇Φ∇UU
for A > 0 and ∆ the Laplacian on the n+ 1 dimensionalmanifold M × R+.
Define a function U(x, y) on M × R+.
U(x, y) = u(x)− 1
2y2∇f∇u
Then∆U = 0
at (x, 0).
Let Φ = |∇ logU |. Then we have
∆Φ ≥ AΦ3 + C∇Φ∇UU
for A > 0 and ∆ the Laplacian on the n+ 1 dimensionalmanifold M × R+.
Define a function U(x, y) on M × R+.
U(x, y) = u(x)− 1
2y2∇f∇u
Then∆U = 0
at (x, 0).Let Φ = |∇ logU |. Then we have
∆Φ ≥ AΦ3 + C∇Φ∇UU
for A > 0 and ∆ the Laplacian on the n+ 1 dimensionalmanifold M × R+.