THE HEAT KERNEL FOR FORMS

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1 THE HEAT KERNEL FOR FORS ATTHEW STEVENSON Abstract. This is the final project for Gantumur Tsogtgerel s partial differential equations class at cgill University. This is a survey of Patodi s construction of the heat kernel for k-forms, as in [4]. We introduce the Riemannian 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 heat kernel. oreover, we discuss the relationship between the heat kernel and the eigenvalues and eigenforms of the Laplace-Beltrami operator. 1. Introduction Let n, g be a compact smooth n-manifold without boundary, where g ΓT T is a Riemannian metric. Let Ω k X := C X, Λ k T be the space of smooth sections of the k-exterior algebra of the cotangent bundle defined over a set X e.g. if X =, Ω k = ΓΛ k T is the space of global differential k-forms. Similarly, let Ω k X := C X, Λ k T Λ k T be the space of smooth double k-forms defined over X. The metric g induces an L -inner product on Ω k by α, β g = g x α, βvol g x, 1.1 for any α, β Ω k, where vol g = detgdx 1... dx n is the standard volume form associated to g. oreover, we can define the Hoe star operator g : Ω k Ω n k by the relation g x α, βvol g x = α g β Λ n T x, 1. for any α, β Ω k and x. It has the property that g = 1 kn k and 1 g = 1 kn k, i.e. the Hoe star is involutive, up to sign. Notice that we can then write α, β g = α gβ. As our metric is fixed, we will often suppress the subscript on the inner product, on the volume form, and on the Hoe star. Let : ΓT T ΓT denote the Levi-Civita connection on, g, which is given on elementary vector fields by the formula x µ = n i,j=1 dx j Γ j iµ xi, 1.3 where the Γ j iµ s denote the Christoffel symbols corresponding to the metric g. By duality, the Levi-Civita connection extends to a map ΓT T ΓT on 1-forms by the formula n dx µ = Γ µ ij dxi dx j. 1.4 i,j=1 Let Tq p denote the space of smooth p, q-tensors on that is, q x i s and p dx j s. These formulae allows us to extend the Levi-Civita connection to a map Tq p T Tp q Tq p+1 by inductively applying the relation Date: April 17, 14. 1

2 ATTHEW STEVENSON ω ζ = ω ζ + ω ζ. In particular, we have an induced map : Λ k T T Λ k T after quotienting T k by symmetric, k-tensors: namely, for α 1,..., α k ΓT, k α 1... α k = 1 i 1 α 1... α i... α k. 1.5 i=1 Let d k : Ω k Ω k+1 denote the exterior derivative on k-forms, and let δ k : Ω k+1 Ω k be the codifferential, which is defined to be the formal adjoint of d k with respect to,. Thus, δ k satisfies d k α, β = α, δ k β for all α Ω k and β Ω k+1. The codifferential can be written explicitly as δ k = 1 nk+1+1 d, in which case it is easy to verify that δ k 1 δ k =, just as d k+1 d k =. The Laplace-Beltrami operator on k-forms is the differential operator k : Ω k Ω k given by k = d k δ k + δ k d k. 1.6 It is an interesting fact, known as the Hoe decomposition theorem, that any smooth k-form can be decomposed into the sum of an exact form, a co-exact form, and a harmonic form. In addition, remark that k is self-adjoint with respect to the L -inner product, because for any α, β Ω k, k α, β = d k δ k α, β δ k d k α, β = δ k α, δ k β d k α, d k β = α, d k δ k β α, δ k d k β = α, k β. Note, as a consequence of the above, that k α β = α k β. Now, the spectral theorem asserts the existence of a complete L -orthonormal subset {ϕ i } i= Ωk of eigenforms of k. In addition, the corresponding eigenvalues can be listed as = λ λ 1... λ i..., where each eigenvalue is repeated according to its finite multiplicity. Let L Ω k be the completion of Ω k with respect to the L -inner product,, then in fact, this set {ϕ i } of eigenforms forms an orthonormal basis of L Ω k. Given ω Ω k, consider the heat equation for k-forms with ω as the initial data: { t k ux, t =, for x, t R +, 1.7 ux, = ωx, for x. Finding a solution ux, t L Ω k R + is equivalent to constructing a fundamental solution of the heat operator on k-forms, by convolving the fundamental solution with ω. A fundamental solution of the heat operator is called a heat kernel; that is, this is a double form 1 e k t, x, y Ω k R + such that t k e k = and for any f Ω k R +, lim e k t, x, y fy, t = fx,. t + This last condition can be reformulated as follows: define the heat propagator e t k : Ω k R + V Ω k V, where V is some coordinate chart, which is given by e t k fy := e k t, x, y x ft, x, 1.8 where x = x 1,..., x n and y = y 1,..., y n are local coordinates in V. ore concretely, if we write the forms in coordinates as e k t, x, y = e I,J t, x, ydx I dy J and ft, x = f K t, xdy K, then e t k fx = g y f I,J t, x, ydy J, f K t, xdy K voly dx I Equivalently, we could consider the heat kernel to be a smooth section of R + Λ k T Λ k T as a vector bundle over the product manifold ; here, R + would be a trivial bundle.

3 THE HEAT KERNEL FOR FORS 3 Therefore, a smooth double form e k t, x, y satisfying t k ye k t, x, y = is the heat kernel if for any ft, x Ω k R +, we have that lim t +e t k fx = f, x, for any x. The construction of a heat kernel will be abetted by the construction of a parametrix for the heat operator, which is a double form G k t, x, y Ω k R + that satisfies the following two conditions: 1 t k yg k t, x, y Ω k R + and t k yg k t, x, y is continuous on R + in time. For any ϕt, x Ω k R +, G k t, x, y ϕt, x = ϕ, y. 1.1 lim t + In??, we construct a parametrix for the heat operator on k-forms, which ultimately leads to the proof of the existence 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 eigenvalues and eigenforms of k. Theorem 1. Assume there exists a heat kernel e k t, x, y, then i= eλit ϕ i x ϕ i y converges locally uniformly on R + to e k t, x, y. Proof. Rosenberg, [5] Fix t R + and x, then et, x, L Ω k and thus can be expressed as a linear combination of Laplace eigenform: indeed, write e k t, x, = i= f it, xϕ i with equality in the L -sense. It follows that f i t, x = ek t, x, y ϕ i y, so consider t f i t, x = t e k t, x, y ϕ i y = k ye k t, x, y ϕ i y = e k t, x, y k yϕ i y = λ i e k t, x, y ϕ i y = λ i f i t, x. Solving this ODE, we find that f i t, x = h i xe λit for some function h i L Ω k. Now, for any f L Ω k, write f = i= a iϕ i for a i R; the second property of the heat kernel implies that for any x, fx = lim e k t, x, y fy t + = lim e λit h i x ϕ i y a i ϕ i y t + i= i= = lim e λjt a j h j x ϕ j y ϕ j y = lim e λjt a j h j x = a j h j x. t + t + j= Therefore, h j x = ϕ j x for all j, and we conclude that e k t, x, y = i= e λit ϕ i x ϕ i y in the L -sense with variable y. Said differently, there exists an increasing sequence of indices {i j } j= such that i j i= j= e λit ϕ i x ϕ i y e k t, x, y as j, where we have pointwise convergence for any t R +, x and for a.e. y. It remains to show that we have pointwise convergence for every y. Remark however that we can write t t t t e λit ϕ i x ϕ i y = e λit/ ϕx e λit/ ϕ i y = e k, x, z z e k, y, z = e k, x,, e k, y,, i= i= j=

4 4 ATTHEW STEVENSON where the second-to-last equality follows from Parseval s identity and the above holds for any t R + and x, y. Since the right-hand side is finite by our earlier considerations, we must have pointwise convergence for all t, x, y. Therefore, we have an increasing sequence { i j i= eλit ϕ i x ϕ i y} j=, which converges pointwise to the heat kernel e k t, x, y for any t R + and x, y. Dini s theorem implies that this convergence is uniform on any compact subset of, 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 []. Corollary. ckean & Singer, 1967 Let dim = n be even, then 1 n 1 k tr 4π n/ x ϕ k n/ x, x volx = χ, 1.11 k= where tr x denotes the trace on Λ k T x Λ k T x.. Construction of a Parametrix Let U be open, and α Ω k U U be a smooth double k-form defined on U U. For a fixed point x, the Riemannian metric g induces an isomorphism Λ k T x [Λ k T x ] given by ω g x ω,, ω Λ k T x. This of course extends to an isomorphism Λ k T [Λ k T ]. Here, V denotes the algebraic dual of V, considered as a vector space. It follows that we can identify Λ k T Λ k T Λ k T [Λ k T ] HomΛ k T, Λ k T. As a consequence, we can consider a smooth double k-form α to be an element of C U U, HomΛ k T, Λ k T. In other words, for each fixed pair x, v U Λ k Tx, we have a map αx, v: U Λ k T. That is, αx, v is a smooth differential k-form defined on U. These two characterizations of double forms will be used interchangeably 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 a small neighbourhood of the diagonal of. Denote the diagonal by diag := {x, x : x }. Indeed, let H k Nt, x, y := e rx,y /4t 4πt d/ N t i U i,k x, y, x, y, t,..1 i= Here, rx, y is the geodesic distance from x to y in and let N > n be some integer. The smooth double forms U i,k C U U, HomΛ k T, Λ k T are defined in a yet to be determined neighbourhood U U of the diagonal, and satisfy the following conditions: 1 For any x U, U,k x, x = id: Λ k T x Λ k T x, the identity morphism. For any x, y U and t,, t k y H k N t, x, y = e rx,y /4t 4πt d/ t N k yu N,k x, y. Here the subscript-y denotes that the Laplace-Beltrami operator acts in the y-variable. Remark that such a double form would satisfy the second condition of a parametrix, as the U i,k s will be the identity sufficiently close to the diagonal. We claim that such double forms U i,k exist and that the above two conditions determine them uniquely. This process is divided into the following two aptly-named subsections.

5 THE HEAT KERNEL FOR FORS 5.1. Existence of the Double Forms. Fix a point x, and consider normal coordinates in an open neighbourhood U of x such that x =,..., and g ij x = δ ij, the Kronecker delta function. Such coordinates exist by Proposition 5.11 of [1], e.g. The open subset x U U will be the open neighbourhood of the diagonal for which the U i,k s are defined. Lemma 3. Let F rx, y be a function of y that is radial with respect to x, in the sense that it only depends on the geodesic distance between the argument y and the fixed point x. Let α C U, Λ p T, then d F y F rα = + n 1 df r + 1 df α + df g r r α + F r α,.3 d where d is differentiating along the geodesic from x to y, and gy = detg ijy. 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 rx, y = e rx,y /4t, then k y e rx,y /4t α = e rx,y /4t rx, y 4t 1 t n 1 t rx, y 4gt α 1 t r α + k α. d Take α = N i= ti U i,k. As a consequence, we can now compute the action of the heat operator on H p N t, x, y: t k yhnt, k x, y /4t N rx, y = e rx,y 4πt n/ 4t + i n/ t i= + t i 1 r d U i,k x, y t i k yu i,k x, y N i + = e rx,y /4t 4πt n/ i= rx, y 4g rx, y 4t + 1 t + n 1 + t rx, y 4gt t i U i,k x, y t i 1 U i,k x, y + t i 1 r d U i,k x, y t i k yu i,k x, y By assumption, we require that only the t N k yu N,k x, y term survives on the right-hand side; in particular, the coefficient of t i 1 must vanish for i =,..., N. Setting this coefficient to zero, we get that rx, y i + U i,k x, y + 4g r d U i,k x, y k yu i 1,k x, y =. Rearranging, we find that for each i =,..., N, i + r d U i,k x, y + rx, y 4g U i,k x, y = k yu i 1,k x, y,.4 where U 1,k x, y :=. Identifying the U i,k s with elements of C U U, HomΛ k T, Λ k T as in the above remark, we have that U i,k v, y Λ k Ty for v Λ k Tx and y U. If we fix v Λ k Tx, then the set of equations given by?? can be reduced to the following system of ODEs in the y-variable: r d U i,k v, y + i + rx, y 4g U i,k v, y = k yu i 1,k v, y.5 where i =,..., N. The problem of constructing a local parametrix then reduces to showing that the system given by?? has a unique solution, and in fact it is unique if we impose the additional constraint that U,k v, x = v. Remark that r d rx, y i g 1/4 U i,k v, y = r d = r i g 1/4 i + r i g 1/4 U i,k v, y + r i g 1/4 r d rx, y 4g U i,k v, y U i,k v, y + rx, y i g 1/4 r d U i,k v, y,

6 6 ATTHEW STEVENSON which means that?? is equivalent to the system rx, y i g 1/4 U i,k v, y = rx, y i g 1/4 k yu i 1,k v, y Λ k Ty,.6 r d for i =,..., N. Fix any y U, and parametrize the geodesic curve u y t between x and y, where t [, rx, y]. This induces a sequence of vector space isomorphism T y,t : Λ k Tu yt Λk Ty more precisely, these isomorphisms are determined by the geodesic and the connection. Consequently, define U,k v, y := gy 1/4 T y, v, then it is clear that U,k v, x = gx 1/4 T x, v = v, as gx = 1 due to the normal coordinate system and T x, is the identity endomorphism on Λ k Tx. We can then construct the other double forms inductively by the formula U i 1,k v, y = 1 rx, y i gy 1/4 rx,y rx, u y t i 1 gu y t 1/4 T y,t k y U i 1,k v, u y t dt.7 Intuitively, this equation is obtained by integrating?? along the geodesic curve from x to y. It is clear that U i,k v, y Λ k Ty and hence U i,k Ω k U U. It remains to verify that these double forms satisfy the system??, but this is actually pretty easy: r d rx, y i g 1/4 U i,k v, y = r d rx,y = rx, y i g 1/4 k yu i 1,k v, y, rx, u y t i 1 gu y t 1/4 T y,t k y U i 1,k v, u y t dt since u y rx, y = y. Therefore, we have established the existence of the double forms U i,k Ω k U U satisfying conditions 1 and... Uniqueness of the Double Forms. The system of equations given by?? can be reduced to iu i,k v, x = k yu i 1,k v, yv, x R,.8 where i =,..., N. Let U i,k 1, U i,k Ω k U U be two sequences of double forms satisfying??, then the above considerations imply that = r d rx, y i g 1/4 U i,k i,k 1 v, y U v, y = rx, yi g 1/4 k yu i 1,k 1 v, y U i 1,k v, y = rx, y i g 1/4 iu i,k 1 It follows that U i,k 1 = U i,k, in other words we have uniqueness of the double forms. i,k v, y U v, y..3. Existence of a Parametrix. In?? and??, we constructed the function HN k t, x, y, which is defined for each x, x diag in some small neighbourhood U U. Obviously, the neighbourhood U depends on x indeed, it comes from the normal coordinates defined at the start of??. Let W = x U U, a small open neighbourhood of diag. Therefore, the function HN k t, x, y is defined for x, y W. Now, let W be another open neighbourhood of diag such that diag W and W intw. Take a non-negative smooth bump function ψ C such that ψ = 1 on W and ψ = on \W. This allows us to define a globally smooth function: G k N t, x, y := ψx, yh k Nt, x, y Ω k R +..9 This is our candidate for the parametrix of the heat operator. By construction, t k yg k N is smooth double form on R + since t k yhn k is as well just apply the product rule, and in fact it is continuous up to zero in time. For ϕt, x Ω k R + that is continuous up to R in time, it remains to show that for any y, G k Nt, x, y x ϕt, x = ϕ, y..1 lim t +

7 THE HEAT KERNEL FOR FORS 7 Figure 1. The open neighbourhoods W, W of diag. To see this, notice that for i =,..., N and > sufficiently small so that B y suppψ W, then e rx,y /4t i,k lim+ ψx, y U x, y x ϕt, x t 4πtn/ e rx,y /4t i,k e rx,y /4t i,k = lim+ U x, y x ϕt, x + lim+ ψx, y U x, y x ϕt, x n/ t t 4πtn/ B y 4πt \B y We will consider these two limits separately. For the second one, notice that e rx,y /4t i,k e /4t ψx, y U x, y ϕt, x ψx, yu i,k x, y x ϕt, x x 4πtn/ 4πtn/ \B y \B y Ce /4t as t 7 +, 4πtn/ where the constant C comes from the integral, which is finite since we are integrating a piecewise-smooth form over the compact subset \B y. For the first integral, we will consider the pullback by the exponential map at y, which gives that e rx,y /4t i,k e rv, /4t i,k U x, y ϕt, x = U expy v, yϕt, expy v detd expy vdv 1... dv n x n/ n/ 4πt 4πt B y B Ty U i,k expy, yϕ, expy detd expy as t 7 +, where U i,k and ϕ are now considered as smooth functions and the last equality follows from the properties of the Euclidean heat kernel on functions. Under the exponential map, this last term is equal to U i,k y, yϕ, y, which denotes the action of U i,k y, y on ϕ, y, considered as an endomorphism of Λk Ty. In summary, we have shown that e rx,y /4t i,k lim+ U x, y x ϕt, x = U i,k y, yϕ, y..11 ψx, y t7 4πtn/ In particular, we have that N X e rx,y /4t i,k lim+ GkN t, x, y x ϕt, x = lim+ ti ψx, y U x, y x ϕt, x = U,k y, yϕ, y = ϕ, y, n/ t t7 4πt i= since U,k y, y is the identity endomorphism on Λk Ty, by assumption. This shows??. Therefore, GkN is indeed a parametrix for the heat operator on k-forms.

8 8 ATTHEW STEVENSON 3. Construction of the Heat Kernel In??, we constructed a global parametrix G k N t, x, y Ωk R + of the heat operator. Consider the image of this parametrix under the heat operator, denoted Kt, x, y := t k yg k N t, x, y, Then for each m 1, inductively define a sequence of double forms by t K m t, x, y := τk m 1 s, x, z, Kt s, z, yvolzds, 3.1 where K := K. Remark that volz = 1 z, and so the above will often be written as such. The candidate for a fundamental solution of the heat operator is given by e k t, x, y := G p N t, x, y + t 1 m+1 τk m s, x, z, G k Nt s, z, yvolzds. 3. m= The terms of this series should be thought of as the convolution of K m and G k N, although these are double forms and so the notion of convolution is not precise. The remaining objectives are then to show that this sum converges, that t k ye k t, x, y = for any t, x, y R +, and finally that it behaves appropriately with respect to the initial data. The latter is obvious, since G k N is a parametrix of the heat operator and the limit of the sum goes to zero. The issue of convergence is where we will start. Take two finite open covers {V i } q i=1, {U i} q i=1 of such that for each i = 1,..., n, V i U i and U i is diffeomorphic to R n. That is, we take any finite open cover {V i } of and nontriviall fatten each V i so that it is diffeomorphic to Euclidean space. Of course, such covers exist as is compact, by assumption. Now, fix a partition of unity {ϕ i } q i=1 subordinate to {V i} q i=1 and fix a sequence of smooth bump functions {ψ i } q i=1 with suppψ i U i and ψ i Vi = 1. The choice of a partition of unity induces a transformation on double forms: for any Lx, y Ω k, define L i,j x, y := ψ i xψ j ylx, y. 3.3 This procedure takes a global double form and just looks at its contribution on U i U j. Similarly, this choice induces a norm on Ω k U i U j in the following manner 3 : write a double form Lx, y Ω k U i U j as Lx, y = α, β =k L α,βx, ydx α dy β, then define L i,j := sup L α,β x, y. 3.4 x U α, β =k i y U j Remark that we have an estimate of the form τα, β i,j C α i,j β i,j for global smooth double forms α and β, where the constant C > depends only on, U i, and U j. Lemma 4. There exists constants, C > such that for all m > and t R +, K m i,jt,, i,j C m+1 t m+1n n/+m ΓN n/ + 1 m+1 Γm + 1N n/ + m Proof. We prove this estimate by induction on m. For the case m = 1, we can use the product rule formula for the Laplace-Beltrami operator to get that t k yψx, yh k N t, x, y = ψx, y t k yh k Nt, x, y + dψ, H k N yψx, yh k N t, x, y. Given 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 τa b, a c := a, a b c, for any a, a A, b B, c C. 3 A remark on notation: the above sum ranges over ordered k-tuples α = α1,..., α k and β = β 1,..., β k in {1,..., n} k such that α 1 <... < α k and β 1 <... < β k. Here, the coefficients are L α,β x, y C U i U j and we let dx α = dx α 1... dx α k and dy β = dy β 1... dy β k respectively, where x 1,..., x n and y 1,..., y n are local coordinates in U i and U j, respectively.

9 THE HEAT KERNEL FOR FORS 9 Here,, does not denote an inner product, but a contraction. 4 Now, let i,j > be the maximum coefficient of the double form k yu N,k on U i U j. Then, Ki,j i,j = ψ i xψ i y t k yψx, yhnt, k x, y i,j n e r /4t i,j sup ψx, y k 4πt n/ tn k yu N,k x, y + Ot N n/ 1 x U i y U j t N n/ + Ot N n/ 1, where absorbs all of the constants, as well as the maximum coefficients of k yu N,k x, y. We can then replace the above with t N n/, by absorbing the Ot N n/ 1 term. Notice that since {ϕ r } is a partition of unity subordinate to the open cover {V r }, then q r=1 ϕ rz = 1 for any z and we can write t Ki,jt, m x, y = τψ i xk m 1 s, x, z, ψ j ykt s, z, yvolzds t q = τψ i x ϕ r z K m 1 s, x, z, ψ j ykt s, z, yvolzds = = q t r=1 t q r=1 r=1 τψ i xϕ r zk m 1 s, x, z, ψ j yψ r zkt s, z, yvolzds V r τψ i xϕ r zk m 1 s, x, z, K r,j t s, z, yvolzds where the second-to-last equality follows since ψ r Vr = 1 and the last integrand is supported in V r, but we will write the integral as being over all of. Now, using the base case, the induction hypothesis, and the above formula all in conjunction, we find that q t Ki,jt, m, i,j τψ i xϕ r zk m 1 s, x, z, K r,j t s, z, y i,i volzds r=1 V i q t C m s mn n/+m 1 ΓN n/ + 1 m ΓmN n/ + m t sn n/ volv i ds r=1 C C m ΓN n/ + 1m ΓmN n/ + m t s mn n/+m 1 t s N n/ ds, where C = q max{volv i : i = 1,..., q}. Take C > C to get the correct powers for the constants. It remains to deal with this last integral. A substitution yields that t 1 s mn n/+m 1 t s N n/ ds = t m+1n n/+m u mn n/+m 1 1 u N n/ du = t m+1n n/+m BetamN n/ + m, N n/ + 1 m+1n n/+m ΓmN n/ + mγn n/ + 1 = t, Γm + 1N n/ + m + 1 where the last equality follows from the identity Betax, y = ΓxΓy/Γx + y. Of course, Beta, denotes the beta function. Substituting this last equality into our previous inequality for Ki,j m t,, i,j gives the desired estimate and completes the induction. 4 For each x, the contraction, x : T x Λk T x T x Λk T x is given by α 1 β α g xα 1, α β. In our case, df x T x and Hk N x Λk T x T. For more details, see page 99 of [5].

10 1 ATTHEW STEVENSON Using the approximation Γλ πλ λ λ e as λ, the lemma gives that K m t, x, y O tam C bm m for m some a, b > and C > constants. However, this gives that the series m= Km t, x, y converges to a double form. It follows that m= 1m+1 K m t, x, y converges to a double form for any t, x, y. Furthermore, the series which defines e k t, x, y can be seen as a convolution of forms as G k N m= 1m+1 K m, and thus this must converge as well. As a consequence, the series defining e k t, x, y converges. It does not immediately follow from our earlier considerations that the partial derivatives of e k t, x, y converge; however, following the same procedure, one may construct estimates of t ν x α y β Ki,j m i,j to get their convergence. It remains to show that e k vanishes under the action of the heat operator. Indeed, remark that t k ye k t, x, y = Kt, x, y + = Kt, x, y + + lim s t = Kt, x, y + = Kt, x, y + m= t 1 m+1 t k y t 1 m+1 m= τk m t, x, z, G k N t s, z, yvolz 1 m+1 K m t, x, y + m= τk m s, x, z, G k Nt s, z, yvolzds t τk m s, x, z, G k N t s, z, yvolzds 1 m+1 K m t, x, y + K m+1 t, x, y m= = Kt, x, y Kt, x, y =. t t k yτk m s, x, z, G k Nt s, z, yvolzds τk m s, x, z, t k yg k Nt s, z, yvolzds Now, let us justify many of the above equalities. The second equality is a direct application of the generalized Leibniz rule; the third equality follows because G k N is a parametrix, and thus the limit picks out the value K m t, x, y. oreover, just by the definition of the map τ we can move the heat operator t k y into one of the coordinates but we cannot move it into both simultaneously. In the fourth equality, we have just replaced the double integral using the inductive definition of K m, as in??. Finally, the sum is telescoping, and the only remaining term is K t, x, y, which cancels with Kt, x, y. It follows that e k t, x, y is indeed in the kernel of the heat operator. Therefore, the double form e k t, x, y given by?? is indeed a heat kernel. References [1] J. Lee, Riemannian Geometry: An Introduction to Curvature. Springer [] H.P. ckean, Jr. & I.. Singer, Curvature and the eigenvalues of the Laplacian. J. Differential Geometry , [3] S. inakshisundaram & A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds. Canadian J. ath , [4] V.K. Patodi, Curvature and the eigenforms of the Laplace operator. J. Differential Geometry , [5] S. Rosenberg, The Laplacian on a Riemannian anifold. Cambrie University Press 1997.