THE HEAT KERNEL FOR FORMS

MATTHEW STEVENSON

Abstract. This is the final project for Gantumur Tsogtgerel’s partial differential equations class at McGill

University. This is a survey of Patodi’s construction of the heat kernel for k-forms, as in [4]. We introduce theRiemannian geometry necessary to frame the heat equation for k-forms on closed manifolds, after which Patodi’s

work is presented. Specifically, we show the existence of a parametrix, which leads to the existence of the heatkernel. Moreover, we discuss the relationship between the heat kernel and the eigenvalues and eigenforms of the

Laplace-Beltrami operator.

1. Introduction

Let (Mn, g) be a compact smooth n-manifold without boundary, where g ∈ Γ(T ∗M ⊗T ∗M) is a Riemannianmetric. Let Ωk(X) := C∞(X,ΛkT ∗M) be the space of smooth sections of the k-exterior algebra of the cotangentbundle defined over a set X (e.g. if X = M , Ωk(M) = Γ(ΛkT ∗M) is the space of global differential k-forms).Similarly, let Ωk2(X) := C∞(X,ΛkT ∗M ⊗ΛkT ∗M) be the space of smooth double k-forms defined over X. Themetric g induces an L2-inner product on Ωk(M) by

〈α, β〉g =

∫M

gx(α, β)volg(x), (1.1)

for any α, β ∈ Ωk, where volg =√

det(g)dx1 ∧ . . .∧ dxn is the standard volume form associated to g. Moreover,

we can define the Hodge star operator ∗g : Ωk(M)→ Ωn−k(M) by the relation

gx(α, β)volg(x) = α ∧ ∗gβ ∈ ΛnT ∗xM, (1.2)

for any α, β ∈ Ωk and x ∈ M . It has the property that ∗2g = (−1)k(n−k) and ∗−1g = (−1)k(n−k), i.e. the Hodge

star is involutive, up to sign. Notice that we can then write 〈α, β〉g =∫Mα ∧ ∗gβ. As our metric is fixed, we

will often suppress the subscript on the inner product, on the volume form, and on the Hodge star.Let ∇ : Γ(TM)→ T ∗M ⊗ Γ(TM) denote the Levi-Civita connection on (M, g), which is given on elementary

vector fields by the formula

∇∂xµ =

n∑i,j=1

dxj ⊗ Γjiµ∂xi , (1.3)

where the Γjiµ’s denote the Christoffel symbols corresponding to the metric g. By duality, the Levi-Civita

connection extends to a map Γ(T ∗M)→ T ∗M ⊗ Γ(T ∗M) on 1-forms by the formula

∇dxµ =

n∑i,j=1

−Γµijdxi ⊗ dxj . (1.4)

Let T pq denote the space of smooth (p, q)-tensors on M (that is, q ∂xi ’s and p dxj ’s). These formulae allows us

to extend the Levi-Civita connection to a map ∇T pq → T ∗M ⊗ T qp ' T p+1q by inductively applying the relation

Date: April 17, 2014.

1

2 MATTHEW STEVENSON

∇(ω ⊗ ζ) = (∇ω)⊗ ζ + ω ⊗ (∇ζ). In particular, we have an induced map ∇ : ΛkT ∗M → T ∗M ⊗ ΛkT ∗M afterquotienting T k0 by symmetric (0, k)-tensors: namely, for α1, . . . , αk ∈ Γ(T ∗M),

∇(α1 ∧ . . . ∧ αk) =

k∑i=1

(−1)i−1α1 ∧ . . . ∧ (∇αi) ∧ . . . ∧ αk. (1.5)

Let dk : Ωk → Ωk+1 denote the exterior derivative on k-forms, and let δk : Ωk+1 → Ωk be the codifferential,which is defined to be the formal adjoint of dk with respect to 〈·, ·〉. Thus, δk satisfies 〈dkα, β〉 = 〈α, δkβ〉 for allα ∈ Ωk and β ∈ Ωk+1. The codifferential can be written explicitly as δk = (−1)n(k+1)+1 ∗ d∗, in which case it iseasy to verify that δk−1δk = 0, just as dk+1dk = 0.

The Laplace-Beltrami operator on k-forms is the differential operator ∆k : Ωk → Ωk given by

∆k = −(dkδk + δkdk). (1.6)

It is an interesting fact, known as the Hodge decomposition theorem, that any smooth k-form can be decomposedinto the sum of an exact form, a co-exact form, and a harmonic form. In addition, remark that ∆k is self-adjointwith respect to the L2-inner product, because for any α, β ∈ Ωk(M),

〈∆kα, β〉 = −〈dkδkα, β〉 − 〈δkdkα, β〉 = −〈δkα, δkβ〉 − 〈dkα, dkβ〉 = −〈α, dkδkβ〉 − 〈α, δkdkβ〉 = 〈α,∆kβ〉.

Note, as a consequence of the above, that (∆kα) ∧ ∗β = α ∧ ∗(∆kβ). Now, the spectral theorem assertsthe existence of a complete L2-orthonormal subset ϕi∞i=0 ⊂ Ωk(M) of eigenforms of ∆k. In addition, thecorresponding eigenvalues can be listed as

0 = λ0 ≥ λ1 ≥ . . . ≥ λi ≥ . . . −∞,

where each eigenvalue is repeated according to its (finite) multiplicity. Let L2Ωk(M) be the completion of Ωk(M)with respect to the L2-inner product 〈·, ·〉, then in fact, this set ϕi of eigenforms forms an orthonormal basisof L2Ωk(M).

Given ω ∈ Ωk(M), consider the heat equation for k-forms with ω as the initial data:(∂t −∆k)u(x, t) = 0, for (x, t) ∈M × R+,

u(x, 0) = ω(x), for x ∈M.(1.7)

Finding a solution u(x, t) ∈ L2Ωk(M × R+) is equivalent to constructing a fundamental solution of the heatoperator on k-forms, by convolving the fundamental solution with ω. A fundamental solution of the heat operatoris called a heat kernel ; that is, this is a double form 1 ek(t, x, y) ∈ Ωk2(R+ ×M ×M) such that (∂t −∆k)ek = 0and for any f ∈ Ωk(M × R+),

limt→0+

∫M

ek(t, x, y) ∧ ∗f(y, t) = f(x, 0).

This last condition can be reformulated as follows: define the heat propagator et∆k

: Ωk(R+ × V ) → Ωk(V ),where V ⊂M is some coordinate chart, which is given by

(et∆k

f)(y) :=

∫M

ek(t, x, y) ∧ ∗xf(t, x), (1.8)

where x = (x1, . . . , xn) and y = (y1, . . . , yn) are local coordinates in V . More concretely, if we write the formsin coordinates as ek(t, x, y) = eI,J(t, x, y)dxI ⊗ dyJ and f(t, x) = fK(t, x)dyK , then

(et∆k

f)(x) =

(∫M

gy(fI,J(t, x, y)dyJ , fK(t, x)dyK)vol(y)

)dxI . (1.9)

1Equivalently, we could consider the heat kernel to be a smooth section of R+ × ΛkT ∗M ⊗ ΛkT ∗M as a vector bundle over theproduct manifold M ×M ; here, R+ would be a trivial M ×M bundle.

THE HEAT KERNEL FOR FORMS 3

Therefore, a smooth double form ek(t, x, y) satisfying (∂t − ∆ky)ek(t, x, y) = 0 is the heat kernel if for any

f(t, x) ∈ Ωk(R+ ×M), we have that limt→0+(et∆k

f)(x) = f(0, x), for any x ∈M .The construction of a heat kernel will be abetted by the construction of a parametrix for the heat operator,

which is a double form Gk(t, x, y) ∈ Ωk2(R+ ×M ×M) that satisfies the following two conditions:

(1) (∂t −∆ky)Gk(t, x, y) ∈ Ωk2(R+ ×M ×M) and (∂t −∆k

y)Gk(t, x, y) is continuous on R+ in time.

(2) For any ϕ(t, x) ∈ Ωk(R+ ×M),

limt→0+

∫M

Gk(t, x, y) ∧ ∗ϕ(t, x) = ϕ(0, y). (1.10)

In ??, we construct a parametrix for the heat operator on k-forms, which ultimately leads to the proof of theexistence of the heat kernel in ??. This construction is due to Patodi, and is originally from his 1971 paper [4].

Below, we explore an amazing relationship between the heat kernel and the spectral data of the Laplace-Beltrami operator. In paticular, we show that the heat kernel on k-forms is locally determined by the eigenvaluesand eigenforms of ∆k.

Theorem 1. Assume there exists a heat kernel ek(t, x, y), then∑∞i=0 e

λitϕi(x)⊗ϕi(y) converges locally uniformlyon R+ ×M ×M to ek(t, x, y).

Proof. (Rosenberg, [5]) Fix t ∈ R+ and x ∈M , then e(t, x, ·) ∈ L2Ωk(M) and thus can be expressed as a linearcombination of Laplace eigenform: indeed, write ek(t, x, ·) =

∑∞i=0 fi(t, x)ϕi(·) with equality in the L2-sense. It

follows that fi(t, x) =∫Mek(t, x, y) ∧ ∗ϕi(y), so consider

∂tfi(t, x) =

∫M

∂tek(t, x, y) ∧ ∗ϕi(y)

=

∫M

∆kyek(t, x, y) ∧ ∗ϕi(y)

=

∫M

ek(t, x, y) ∧ ∗∆kyϕi(y) = λi

∫M

ek(t, x, y) ∧ ∗ϕi(y) = λifi(t, x).

Solving this ODE, we find that fi(t, x) = hi(x)eλit for some function hi ∈ L2Ωk. Now, for any f ∈ L2Ωk, writef =

∑∞i=0 aiϕi for ai ∈ R; the second property of the heat kernel implies that for any x ∈M ,

f(x) = limt→0+

∫M

ek(t, x, y) ∧ ∗f(y)

= limt→0+

∫M

( ∞∑i=0

eλithi(x)⊗ ϕi(y)

)∧ ∗

( ∞∑i=0

aiϕi(y)

)

= limt→0+

∫M

∞∑j=0

∫M

eλjtajhj(x)⊗ ϕj(y) ∧ ∗ϕj(y) = limt→0+

∞∑j=0

eλjtajhj(x) =

∞∑j=0

ajhj(x).

Therefore, hj(x) = ϕj(x) for all j, and we conclude that ek(t, x, y) =∑∞i=0 e

−λitϕi(x) ⊗ ϕi(y) in the L2-sensewith variable y. Said differently, there exists an increasing sequence of indices ij∞j=0 such that

ij∑i=0

eλitϕi(x)⊗ ϕi(y)→ ek(t, x, y) as j →∞,

where we have pointwise convergence for any t ∈ R+, x ∈ M and for a.e. y ∈ M . It remains to show that wehave pointwise convergence for every y ∈M . Remark however that we can write∞∑i=0

eλitϕi(x)⊗ϕi(y) =

∞∑i=0

e−λit/2ϕ(x)⊗e−λit/2ϕi(y) =

∫M

ek(t

2, x, z

)∧∗zek

(t

2, y, z

)= 〈ek

(t

2, x, ·

), ek(t

2, y, ·

)〉,

4 MATTHEW STEVENSON

where the second-to-last equality follows from Parseval’s identity and the above holds for any t ∈ R+ andx, y ∈ M . Since the right-hand side is finite by our earlier considerations, we must have pointwise convergencefor all t, x, y.

Therefore, we have an increasing sequence ∑iji=0 e

λitϕi(x) ⊗ ϕi(y)∞j=0, which converges pointwise to the

heat kernel ek(t, x, y) for any t ∈ R+ and x, y ∈ M . Dini’s theorem implies that this convergence is uniform onany compact subset of M , and hence we have the required locally uniform convergence.

If ϕ(k)i denotes the sequence of eigenforms of ∆k, then the construction of the heat kernel yields the following

corollary. A proof of this result (along with similar results) is given in [5], and the original is from [2].

Corollary 2. (McKean & Singer, 1967) Let dim(M) = n be even, then

1

(4π)n/2

n∑k=0

(−1)k∫M

trx

(ϕ

(k)n/2(x, x)

)vol(x) = χ(M), (1.11)

where trx denotes the trace on ΛkT ∗xM ⊗ ΛkT ∗xM .

2. Construction of a Parametrix

Let U ⊂ M be open, and α ∈ Ωk2(U × U) be a smooth double k-form defined on U × U . For a fixed pointx ∈M , the Riemannian metric g induces an isomorphism ΛkT ∗xM → [ΛkT ∗xM ]′ given by

ω 7→ gx(ω, ·), ω ∈ ΛkT ∗xM.

This of course extends to an isomorphism ΛkT ∗M∼→ [ΛkT ∗M ]′. Here, V ′ denotes the algebraic dual of V ,

considered as a vector space. It follows that we can identify

ΛkT ∗M ⊗ ΛkT ∗M ' ΛkT ∗M ⊗ [ΛkT ∗M ]′ ' Hom(ΛkT ∗M,ΛkT ∗M).

As a consequence, we can consider a smooth double k-form α to be an element of C∞(U×U,Hom(ΛkT ∗M,ΛkT ∗M)).In other words, for each fixed pair (x, v) ∈ U × ΛkT ∗xM , we have a map α(x, •)(v) : U → ΛkT ∗•M . That is,α(x, •)(v) is a smooth differential k-form defined on U . These two characterizations of double forms will be usedinterchangeably in our work below.

The goal of this section is to construct a local parametrix for ∂t −∆k, in the sense that it is only valid in asmall neighbourhood of the diagonal of M ×M . Denote the diagonal by diag(M) := (x, x) ∈M ×M : x ∈M.Indeed, let

HkN (t, x, y) :=

e−r(x,y)2/4t

(4πt)d/2

(N∑i=0

ti · U i,k(x, y)

), x, y ∈M, t ∈ (0,∞). (2.1)

Here, r(x, y) is the geodesic distance from x to y in M and let N > n2 be some integer. The smooth double

forms U i,k ∈ C∞(U ×U,Hom(ΛkT ∗M,ΛkT ∗M)) are defined in a yet to be determined neighbourhood U ×U ofthe diagonal, and satisfy the following conditions:

(1) For any x ∈ U , U0,k(x, x) = id: ΛkT ∗xM → ΛkT ∗xM , the identity morphism.(2) For any x, y ∈ U and t ∈ (0,∞),(

∂t −∆ky

)HkN (t, x, y) = −e

−r(x,y)2/4t

(4πt)d/2tN ·∆k

yUN,k(x, y) (2.2)

Here the subscript-y denotes that the Laplace-Beltrami operator acts in the y-variable. Remark that sucha double form would satisfy the second condition of a parametrix, as the U i,k’s will be the identity sufficientlyclose to the diagonal. We claim that such double forms U i,k exist and that the above two conditions determinethem uniquely. This process is divided into the following two aptly-named subsections.

THE HEAT KERNEL FOR FORMS 5

2.1. Existence of the Double Forms. Fix a point x ∈ M , and consider normal coordinates in an openneighbourhood U of x such that x = (0, . . . , 0) and gij(x) = δij , the Kronecker delta function. Such coordinatesexist by Proposition 5.11 of [1], e.g. The open subset ∪x∈MU×U will be the open neighbourhood of the diagonalfor which the U i,k’s are defined.

Lemma 3. Let F (r(x, y)) be a function of y ∈ M that is radial (with respect to x), in the sense that it onlydepends on the geodesic distance between the argument y and the fixed point x. Let α ∈ C∞(U,ΛpT ∗M), then

∆y(F (r)α) =

(d2F

dr2+n− 1

r

dF

dr+

1

2g

dg

dr

dF

dr

)α+

2

r

dF

dr∇r ddrα+ F (r)∆α, (2.3)

where ddr is differentiating along the geodesic from x to y, and g(y) = det(gij(y)).

Proof. This is an application of the ‘product rule’ for the Laplace-Beltrami operator (see page 99 of [5]).

Let us apply the above Lemma to F (r(x, y)) = e−r(x,y)2/4t, then

∆ky

(e−r(x,y)2/4tα

)= e−r(x,y)2/4t

((r(x, y)2

4t2− 1

2t− n− 1

2t− r(x, y)

4gt

dg

dr

)α− 1

t∇r ddrα+ ∆kα

).

Take α =∑Ni=0 t

iU i,k. As a consequence, we can now compute the action of the heat operator on HpN (t, x, y):

(∂t −∆ky)Hk

N (t, x, y)

=e−r(x,y)2/4t

(4πt)n/2

N∑i=0

((r(x, y)2

4t2+i− n/2

t− r(x, y)2

4t2+

1

2t+n− 1

2t+r(x, y)

4gt

dg

dr

)tiU i,k(x, y)

+ ti−1∇r ddrUi,k(x, y)− ti∆k

yUi,k(x, y)

)=e−r(x,y)2/4t

(4πt)n/2

N∑i=0

((i+

r(x, y)

4g

dg

dr

)ti−1U i,k(x, y) + ti−1∇r ddrU

i,k(x, y)− ti∆kyU

i,k(x, y)

)By assumption, we require that only the tN∆k

yUN,k(x, y) term survives on the right-hand side; in particular, the

coefficient of ti−1 must vanish for i = 0, . . . , N . Setting this coefficient to zero, we get that(i+

r(x, y)

4g

dg

dr

)U i,k(x, y) +∇r ddrU

i,k(x, y)−∆kyU

i−1,k(x, y) = 0.

Rearranging, we find that for each i = 0, . . . , N ,

∇r ddrUi,k(x, y) +

(i+

r(x, y)

4g

dg

dr

)U i,k(x, y) = ∆k

yUi−1,k(x, y), (2.4)

where U−1,k(x, y) := 0. Identifying the U i,k’s with elements of C∞(U × U,Hom(ΛkT ∗M,ΛkT ∗M)) as in theabove remark, we have that U i,k(v, y) ∈ ΛkT ∗yM for v ∈ ΛkT ∗xM and y ∈ U . If we fix v ∈ ΛkT ∗xM , then the setof equations given by ?? can be reduced to the following system of ODEs in the y-variable:

∇r ddrUi,k(v, y) +

(i+

r(x, y)

4g

dg

dr

)U i,k(v, y) = ∆k

yUi−1,k(v, y) (2.5)

where i = 0, . . . , N . The problem of constructing a local parametrix then reduces to showing that the systemgiven by ?? has a unique solution, and in fact it is unique if we impose the additional constraint that U0,k(v, x) =v. Remark that

∇r ddr(r(x, y)ig1/4U i,k(v, y)

)= r

d

dr

(rig1/4

)U i,k(v, y) + rig1/4∇r ddrU

i,k(v, y)

= rig1/4

(i+

r(x, y)

4g

dg

dr

)U i,k(v, y) + r(x, y)ig1/4∇r ddrU

i,k(v, y),

6 MATTHEW STEVENSON

which means that ?? is equivalent to the system

∇r ddr(r(x, y)ig1/4U i,k(v, y)

)= r(x, y)ig1/4∆k

yUi−1,k(v, y) ∈ ΛkT ∗yM, (2.6)

for i = 0, . . . , N . Fix any y ∈ U , and parametrize the geodesic curve uy(t) between x and y, where t ∈[0, r(x, y)]. This induces a sequence of vector space isomorphism Ty,t0 : ΛkT ∗uy(t0)M → ΛkT ∗yM (more precisely,

these isomorphisms are determined by the geodesic and the connection).Consequently, define U0,k(v, y) := g(y)1/4Ty,0(v), then it is clear that U0,k(v, x) = g(x)1/4Tx,0(v) = v, as

g(x) = 1 due to the normal coordinate system and Tx,0 is the identity endomorphism on ΛkT ∗xM . We can thenconstruct the other double forms inductively by the formula

U i−1,k(v, y) =1

r(x, y)ig(y)1/4

∫ r(x,y)

0

r(x, uy(t))i−1g(uy(t))1/4Ty,t(∆kyU

i−1,k(v, uy(t)))dt (2.7)

Intuitively, this equation is obtained by integrating ?? along the geodesic curve from x to y. It is clear thatU i,k(v, y) ∈ ΛkT ∗yM and hence U i,k ∈ Ωk2(U×U). It remains to verify that these double forms satisfy the system??, but this is actually pretty easy:

∇r ddr(r(x, y)ig1/4U i,k(v, y)

)= ∇r ddr

∫ r(x,y)

0

r(x, uy(t))i−1g(uy(t))1/4Ty,t(∆kyU

i−1,k(v, uy(t)))dt

= r(x, y)ig1/4∆kyU

i−1,k(v, y),

since uy(r(x, y)) = y. Therefore, we have established the existence of the double forms U i,k ∈ Ω2k(U × U)

satisfying conditions (1) and (2).

2.2. Uniqueness of the Double Forms. The system of equations given by ?? can be reduced to

iU i,k(v, x) = (∆kyU

i−1,k(v, y))(v, x) ∈ R, (2.8)

where i = 0, . . . , N . Let U i,k1 , U i,k2 ∈ Ωk2(U ×U) be two sequences of double forms satisfying ?? , then the aboveconsiderations imply that

0 = ∇r ddr (r(x, y)ig1/4(U i,k1 (v, y)− U i,k2 (v, y))) = r(x, y)ig1/4∆ky(U i−1,k

1 (v, y)− U i−1,k2 (v, y))

= r(x, y)ig1/4i(U i,k1 (v, y)− U i,k2 (v, y)).

It follows that U i,k1 = U i,k2 , in other words we have uniqueness of the double forms.

2.3. Existence of a Parametrix. In ?? and ??, we constructed the function HkN (t, x, y), which is defined

for each (x, x) ∈ diag(M) in some small neighbourhood U × U ⊂ M ×M . Obviously, the neighbourhood Udepends on x (indeed, it comes from the normal coordinates defined at the start of ??). Let W = ∪x∈MU ×U , asmall open neighbourhood of diag(M) ⊂M ×M . Therefore, the function Hk

N (t, x, y) is defined for (x, y) ∈ W .

Now, let W ′ be another open neighbourhood of diag(M) such that diag(M) ( W ′ and W ′ ⊂ int(W ). Take anon-negative smooth bump function ψ ∈ C∞(M ×M) such that ψ = 1 on W ′ and ψ = 0 on (M ×M)\W . Thisallows us to define a globally smooth function:

GkN (t, x, y) := ψ(x, y)HkN (t, x, y) ∈ Ωk2(R+ ×M ×M). (2.9)

This is our candidate for the parametrix of the heat operator. By construction, (∂t −∆ky)GkN is smooth double

form on R+×M ×M since (∂t−∆ky)Hk

N is as well (just apply the product rule), and in fact it is continuous upto zero in time.

For ϕ(t, x) ∈ Ωk(R+ ×M) that is continuous up to 0 ∈ R in time, it remains to show that for any y ∈M ,

limt→0+

∫M

GkN (t, x, y) ∧ ∗xϕ(t, x) = ϕ(0, y). (2.10)

THE HEAT KERNEL FOR FORMS 7

Figure 1. The open neighbourhoods W,W ′ of diag(M).

To see this, notice that for i = 0, . . . , N and ε > 0 sufficiently small so that Bε(y) ⊂ supp(ψ) ⊂W ′, then

limt→0+

∫M

ψ(x, y)e−r(x,y)2/4t

(4πt)n/2U i,k(x, y) ∧ ∗xϕ(t, x)

= limt→0+

∫Bε(y)

e−r(x,y)2/4t

(4πt)n/2U i,k(x, y) ∧ ∗xϕ(t, x) + lim

t→0+

∫M\Bε(y)

ψ(x, y)e−r(x,y)2/4t

(4πt)n/2U i,k(x, y) ∧ ∗xϕ(t, x)

We will consider these two limits separately. For the second one, notice that∣∣∣∣∣∫M\Bε(y)

ψ(x, y)e−r(x,y)2/4t

(4πt)n/2U i,k(x, y) ∧ ∗xϕ(t, x)

∣∣∣∣∣ ≤ e−ε2/4t

(4πt)n/2

∫M\Bε(y)

|ψ(x, y)U i,k(x, y) ∧ ∗xϕ(t, x)|

≤ Ce−ε2/4t

(4πt)n/2→ 0 as t 7→ 0+,

where the constant C comes from the integral, which is finite since we are integrating a piecewise-smooth formover the compact subset M\Bε(y). For the first integral, we will consider the pullback by the exponential mapat y, which gives that∫Bε(y)

e−r(x,y)2/4t

(4πt)n/2U i,k(x, y) ∧ ∗xϕ(t, x) =

∫Bε(0)⊂TyM

e−r(v,0)2/4t

(4πt)n/2U i,k(expy v, y)ϕ(t, expy v) det(d expy)(v)dv1 . . . dvn

→ U i,k(expy 0, y)ϕ(0, expy 0) det(d expy)(0) as t 7→ 0+,

where U i,k and ϕ are now considered as smooth functions and the last equality follows from the properties ofthe Euclidean heat kernel on functions. Under the exponential map, this last term is equal to U i,k(y, y)ϕ(0, y),which denotes the action of U i,k(y, y) on ϕ(0, y), considered as an endomorphism of ΛkT ∗yM . In summary, wehave shown that

limt7→0+

∫M

ψ(x, y)e−r(x,y)2/4t

(4πt)n/2U i,k(x, y) ∧ ∗xϕ(t, x) = U i,k(y, y)ϕ(0, y). (2.11)

In particular, we have that

limt→0+

∫M

GkN (t, x, y)∧∗xϕ(t, x) =

N∑i=0

limt7→0+

ti∫M

ψ(x, y)e−r(x,y)2/4t

(4πt)n/2U i,k(x, y)∧∗xϕ(t, x) = U0,k(y, y)ϕ(0, y) = ϕ(0, y),

since U0,k(y, y) is the identity endomorphism on ΛkT ∗yM , by assumption. This shows ??. Therefore, GkN isindeed a parametrix for the heat operator on k-forms.

8 MATTHEW STEVENSON

3. Construction of the Heat Kernel

In ??, we constructed a global parametrix GkN (t, x, y) ∈ Ωk2(R+ ×M ×M) of the heat operator. Considerthe image of this parametrix under the heat operator, denoted K(t, x, y) := (∂t −∆k

y)GkN (t, x, y), Then for each

m ≥ 1, inductively define a sequence of double forms by2

Km(t, x, y) :=

∫ t

0

∫M

τ(Km−1(s, x, z),K(t− s, z, y))vol(z)ds, (3.1)

where K0 := K. Remark that vol(z) = ∗1z, and so the above will often be written as such. The candidate for afundamental solution of the heat operator is given by

ek(t, x, y) := GpN (t, x, y) +

∞∑m=0

(−1)m+1

∫ t

0

∫M

τ(Km(s, x, z), GkN (t− s, z, y))vol(z)ds. (3.2)

The terms of this series should be thought of as the convolution of Km and GkN , although these are doubleforms and so the notion of convolution is not precise. The remaining objectives are then to show that this sumconverges, that (∂t−∆k

y)ek(t, x, y) = 0 for any (t, x, y) ∈ R+×M ×M , and finally that it behaves appropriately

with respect to the initial data. The latter is obvious, since GkN is a parametrix of the heat operator and thelimit of the sum goes to zero. The issue of convergence is where we will start.

Take two finite open covers Viqi=1, Uiqi=1 of M such that for each i = 1, . . . , n, Vi ⊂ Ui and Ui is diffeo-

morphic to Rn. That is, we take any finite open cover Vi of M and nontriviall ‘fatten’ each Vi so that it isdiffeomorphic to Euclidean space. Of course, such covers exist as M is compact, by assumption.

Now, fix a partition of unity ϕiqi=1 subordinate to Viqi=1 and fix a sequence of smooth bump functionsψiqi=1 with supp(ψi) ⊆ Ui and ψi|Vi = 1. The choice of a partition of unity induces a transformation on doubleforms: for any L(x, y) ∈ Ωk2(M ×M), define

Li,j(x, y) := ψi(x)ψj(y)L(x, y). (3.3)

This procedure takes a global double form and just looks at its contribution on Ui × Uj . Similarly, this choiceinduces a norm on Ωk2(Ui × Uj) in the following manner3: write a double form L(x, y) ∈ Ωk2(Ui × Uj) asL(x, y) =

∑|α|,|β|=k Lα,β(x, y)(dx)α ⊗ (dy)β , then define

‖L‖i,j :=∑

|α|,|β|=k

supx∈Uiy∈Uj

|Lα,β(x, y)|. (3.4)

Remark that we have an estimate of the form ‖τ(α, β)‖i,j ≤ C‖α‖i,j‖β‖i,j for global smooth double forms αand β, where the constant C > 0 depends only on M,Ui, and Uj .

Lemma 4. There exists constants M,C > 0 such that for all m ∈ Z>0 and t ∈ R+,

‖Kmi,j(t, ·, ·)‖i,j ≤ (CM)m+1t(m+1)(N−n/2)+m Γ(N − n/2 + 1)m+1

Γ((m+ 1)(N − n/2) +m+ 1). (3.5)

Proof. We prove this estimate by induction on m. For the case m = 1, we can use the ‘product rule’ formulafor the Laplace-Beltrami operator to get that

(∂t −∆ky)ψ(x, y)Hk

N (t, x, y) = ψ(x, y)(∂t −∆ky)Hk

N (t, x, y) + 2〈dψ,∇HkN 〉 − (∆0

yψ(x, y))HkN (t, x, y).

2Given vector spaces A,B,C over a common base field where A is equipped with an inner product, there is a ‘projection’-type

linear map on the product of the tensor product spaces, denoted τ : (A⊗B)× (A⊗ C)→ B ⊗ C. This is given explicitly by

τ(a⊗ b, a′ ⊗ c) := 〈a, a′〉 · b⊗ c,for any a, a′ ∈ A, b ∈ B, c ∈ C.

3A remark on notation: the above sum ranges over ordered k-tuples α = (α1, . . . , αk) and β = (β1, . . . , βk) in 1, . . . , nk suchthat α1 < . . . < αk and β1 < . . . < βk. Here, the coefficients are Lα,β(x, y) ∈ C∞(Ui × Uj) and we let (dx)α = dxα1 ∧ . . . ∧ dxαkand (dy)β = dyβ1 ∧ . . . ∧ dyβk respectively, where x1, . . . , xn and y1, . . . , yn are local coordinates in Ui and Uj , respectively.

THE HEAT KERNEL FOR FORMS 9

Here, 〈·, ·〉 does not denote an inner product, but a contraction.4Now, let Mi,j > 0 be the maximum coefficientof the double form ∆k

yUN,k on Ui × Uj . Then,

‖K0i,j‖i,j = ‖ψi(x)ψi(y)(∂t −∆k

y)(ψ(x, y)HkN (t, x, y))‖i,j

≤(n

k

)Mi,j sup

x∈Uiy∈Uj

∣∣∣∣∣ψ(x, y)e−r

2/4t

(4πt)n/2tN∆k

yUN,k(x, y) +O(tN−n/2−1)

∣∣∣∣∣≤ M ′tN−n/2 +O(tN−n/2−1),

where M ′ absorbs all of the constants, as well as the maximum coefficients of ∆kyU

N,k(x, y). We can then replace

the above with ≤MtN−n/2, by absorbing the O(tN−n/2−1) term. Notice that since ϕr is a partition of unitysubordinate to the open cover Vr, then

∑qr=1 ϕr(z) = 1 for any z ∈M and we can write

Kmi,j(t, x, y) =

∫ t

0

∫M

τ(ψi(x)Km−1(s, x, z), ψj(y)K(t− s, z, y))vol(z)ds

=

∫ t

0

∫M

τ(ψi(x)

(q∑r=1

ϕr(z)

)Km−1(s, x, z), ψj(y)K(t− s, z, y))vol(z)ds

=

q∑r=1

∫ t

0

∫Vr

τ(ψi(x)ϕr(z)Km−1(s, x, z), ψj(y)ψr(z)K(t− s, z, y))vol(z)ds

=

q∑r=1

∫ t

0

∫M

τ(ψi(x)ϕr(z)Km−1(s, x, z),Kr,j(t− s, z, y))vol(z)ds

where the second-to-last equality follows since ψr|Vr = 1 and the last integrand is supported in Vr, but we willwrite the integral as being over all of M . Now, using the base case, the induction hypothesis, and the aboveformula all in conjunction, we find that

‖Kmi,j(t, ·, ·)‖i,j ≤

q∑r=1

∫ t

0

∫Vi

‖τ(ψi(x)ϕr(z)Km−1(s, x, z),Kr,j(t− s, z, y))‖i,ivol(z)ds

≤q∑r=1

∫ t

0

(CM)msm(N−n/2)+m−1 Γ(N − n/2 + 1)m

Γ(m(N − n/2) +m)·M(t− s)N−n/2 · vol(Vi)ds

≤ C ′ · (CM)mMΓ(N − n/2 + 1)m

Γ(m(N − n/2) +m)

∫ t

0

sm(N−n/2)+m−1(t− s)N−n/2ds,

where C ′ = q ·maxvol(Vi) : i = 1, . . . , q. Take C > C ′ to get the correct powers for the constants. It remainsto deal with this last integral. A substitution yields that∫ t

0

sm(N−n/2)+m−1(t− s)N−n/2ds = t(m+1)(N−n/2)+m

∫ 1

0

um(N−n/2)+m−1(1− u)N−n/2du

= t(m+1)(N−n/2)+m · Beta(m(N − n/2) +m,N − n/2 + 1)

= t(m+1)(N−n/2)+mΓ(m(N − n/2) +m)Γ(N − n/2 + 1)

Γ((m+ 1)(N − n/2) + (m+ 1)),

where the last equality follows from the identity Beta(x, y) = Γ(x)Γ(y)/Γ(x + y). Of course, Beta(·, ·) denotesthe beta function. Substituting this last equality into our previous inequality for ‖Km

i,j(t, ·, ·)‖i,j gives the desiredestimate and completes the induction.

4For each x ∈M , the contraction 〈·, ·〉x : T ∗xM ⊗ (ΛkT ∗xM ⊗ T ∗xM)→ ΛkT ∗xM is given by α1 ⊗ (β ⊗ α2) 7→ gx(α1, α2)β. In ourcase, (df)x ∈ T ∗xM and (∇Hk

N )x ∈ ΛkT ∗xM ⊗ T ∗M . For more details, see page 99 of [5].

10 MATTHEW STEVENSON

Using the approximation Γ(λ) ∼√

2πλ(λe

)λas λ→∞, the lemma gives that |Km(t, x, y)| ≤ O( t

amCbm

mm ) for

some a, b > 0 and C > 0 constants. However, this gives that the series∑∞m=0K

m(t, x, y) converges to a doubleform. It follows that

∑∞m=0(−1)m+1Km(t, x, y) converges to a double form for any t, x, y. Furthermore, the

series which defines ek(t, x, y) can be seen as a ‘convolution of forms’ as GkN ∗∑∞m=0(−1)m+1Km, and thus this

must converge as well. As a consequence, the series defining ek(t, x, y) converges.It does not immediately follow from our earlier considerations that the partial derivatives of ek(t, x, y) converge;

however, following the same procedure, one may construct estimates of ‖∂νt ∂αx ∂βyKmi,j‖i,j to get their convergence.

It remains to show that ek vanishes under the action of the heat operator. Indeed, remark that

(∂t −∆ky)ek(t, x, y)

= K(t, x, y) +

∞∑m=0

(−1)m+1(∂t −∆ky)

t∫0

∫M

τ(Km(s, x, z), GkN (t− s, z, y))vol(z)ds

= K(t, x, y) +

∞∑m=0

(−1)m+1

(∫ t

0

∫M

∂tτ(Km(s, x, z), GkN (t− s, z, y))vol(z)ds

+ lims→t−

∫M

τ(Km(t, x, z), GkN (t− s, z, y))vol(z)

t∫0

∫M

∆kyτ(Km(s, x, z), GkN (t− s, z, y))vol(z)ds

)

= K(t, x, y) +

∞∑m=0

(−1)m+1

Km(t, x, y) +

t∫0

∫M

τ(Km(s, x, z), (∂t −∆ky)GkN (t− s, z, y))vol(z)ds

= K(t, x, y) +

∞∑m=0

(−1)m+1(Km(t, x, y) +Km+1(t, x, y)

)= K(t, x, y)−K(t, x, y) = 0.

Now, let us justify many of the above equalities. The second equality is a direct application of the generalizedLeibniz rule; the third equality follows because GkN is a parametrix, and thus the limit picks out the valueKm(t, x, y). Moreover, just by the definition of the map τ we can move the heat operator (∂t −∆k

y) into one ofthe coordinates (but we cannot move it into both simultaneously). In the fourth equality, we have just replacedthe double integral using the inductive definition of Km, as in ??. Finally, the sum is telescoping, and the onlyremaining term is −K0(t, x, y), which cancels with K(t, x, y). It follows that ek(t, x, y) is indeed in the kernel ofthe heat operator.

Therefore, the double form ek(t, x, y) given by ?? is indeed a heat kernel.

References

[1] J. Lee, Riemannian Geometry: An Introduction to Curvature. Springer (1997).[2] H.P. McKean, Jr. & I.M. Singer, Curvature and the eigenvalues of the Laplacian. J. Differential Geometry 1 (1967), 43-69.

[3] S. Minakshisundaram & A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds.

Canadian J. Math. 1 (1949), 242-256.[4] V.K. Patodi, Curvature and the eigenforms of the Laplace operator. J. Differential Geometry 5 (1971), 233-249.[5] S. Rosenberg, The Laplacian on a Riemannian Manifold. Cambridge University Press (1997).