Strong short-time asymptotics and convolution approximation of the heat kernel

We give a short proof of a strong version of the short-time asymptotic expansion of heat kernels associated with Laplace-type operators acting on sections of vector bundles over compact Riemannian manifolds, including exponential decay of the difference of the approximate heat kernel and the true heat kernel. We use this to show that repeated convolution of the approximate heat kernels can be used to approximate the heat kernel on all of M, which is related to expressing the heat kernel as a path integral. This scheme is then applied to obtain a short-time asymptotic expansion of the heat kernel at the cut locus.


Introduction and main results
Let M be a compact Riemannian manifold of dimension n and let L be a Laplace-type operator, acting on sections of a vector bundle V over M. For t > 0, the heat kernel p L t of L is a smooth section of the bundle V V * over M × M (the vector bundle with fiber Hom(V y , V x ) over the point (x, y) ∈ M × M). It is well known that for x, y ∈ M close, the heat kernel has an asymptotic expansion of the form such that there is a unique minimizing geodesic connecting x and y (compare, e.g., [7,Section 2.5]). In this paper, we will prove that the asymptotic relation (1.1) can be made precise as follows.

Theorem 1.1 (Strong heat kernel asymptotics) Let L be a Laplace-type operator, acting on sections of a vector bundle V over a compact Riemannian manifold M. Then for any compact subset K of M
M, any T > 0 and any numbers ν, k, l, m ∈ N 0 , there exists a constant C > 0 such that for all (x, y) ∈ K , whenever 0 < t ≤ T . Here j (x, y) are the heat kernel coefficients as discussed above.
In the theorem, ∇ x and ∇ y denote the covariant derivative with respect to the x (respectively y) variable, where we use any metric connection on the bundle V (changing the connection only alters the constant C on the right-hand side). Usually, the asymptotic relation (1.1) is interpreted to say that for any ν ∈ N 0 , p L t (x, y) − e t (x, y) ν j=0 t j j (x, y) j! ≤ Ct ν+1 (1.4) uniformly for (x, y) over compact subsets of M M. This statement is much weaker than Thm. 1.1 (even in the case that k = l = m = 0), since the latter implies that the right-hand side of (1.4) can be replaced by Ct ν+1 e t (x, y), which decays exponentially when d(x, y) > 0. Proofs for the weaker statement can be found in various places in the literature (see [7,Thm. 2.30], [30,Thm. 7.15], [31, 3.2], [5,III.E] just to name a few 1 ). The stronger result of Thm. 1.1 seems to be somewhat folklore, but to the author's knowledge, no easily accessible proof exists in the literature outside either the theory of pseudo-differential operators, where one usually proves more general statements using a somewhat huge machinery (see, e.g., [17,24]), or the realm of stochastic analysis (e.g., [2,3,26]).
To illustrate the power of these results, we note that an easy corollary is the following result on the symmetry of heat kernel coefficients (compare Corollary 3.3), which is not at all obvious from the defining equations of the j and was previously proved using substantially more involved arguments (see Remark 3.4).

Theorem 1.3 The heat kernel coefficients satisfy the symmetry relation
where * j are the heat kernel coefficients for the heat kernel p * t of the formally adjoint operator L * and ( * j (y, x)) * denotes the fiberwise metric adjoint of * j (y, x).
The first goal of this paper is to give an easy proof of Thm. 1.1 using the so-called transmutation formula, which relates the heat equation to the wave equation, and the Hadamard expansion of the wave kernel. This approach goes back to an older paper of Kannai [22], who proves a variant of Thm. 1.1 in the scalar case (compare also [34]).
Thm. 1.1 can be generalized to general complete manifolds. However, this is a somewhat intricate matter, as general Laplace-type operators need not have closed extensions generating operator semigroups. For formally self-adjoint Laplace-type operators L, we prove that they have at most one such self-adjoint extension and that if they do, a version of Thm. 1.1 holds for the corresponding heat kernel.
The asymptotic expansion (1.1) suggests to define approximate heat kernels e ν t (x, y) by is a smooth function with χ(r ) = 1 near zero and support contained in [0, inj(M)) (with inj(M) denoting the injectivity radius of M). If for general smooth kernels k, it turns out that the heat kernel p L t (x, y) can be approximated by repeated convolutions of the kernel e ν t (x, y). More precisely, we have the following result. any ν ∈ N 0 and each T > 0, there exists a constant C > 0 such that for all x, y ∈ M and for any partition where p t is the heat kernel of the Laplace-Beltrami operator on M. Here we used the notations j τ := τ j − τ j−1 and |τ | := max 1≤ j≤N j τ for the increment, respectively the mesh of a partition τ .
This approximation result can be used in different regimes: If one fixes t > 0, one can make the C k difference in between p L t and e ν 1 τ * · · · * e ν N τ smaller than any given ε > 0, by choosing a partition τ fine enough. On the other hand, by choosing ν large enough, this error can be made uniform in t.
The estimate from Thm. 1.4 is an a posteriori estimate, in the sense that the error depends on p t (x, y), which itself is the (a priori unknown) solution to a differential equation. One can obtain an a priori estimate by using the Gaussian estimate from above [20,Thm. 5.3.4], p t (x, y) ≤ Ct −n/2+1/2 e t (x, y), which holds on compact Riemannian manifolds: One gets that one can replace the result of Thm. 1.4 by the estimate This is a weaker statement, however, since, for example, if (x, y) ∈ M M, one even has p t (x, y) ≤ Ce t (x, y) for all 0 < t ≤ T , so in this case, the additional factor of t −n/2 can be dropped on the right-hand side of (1.8) (with the constant being uniform over compact subsets of M M in this case). Similar approximation schemes and their relation to finite-dimensional approximation of path integrals have also been considered by Fine and Sawin, who use these to give a "path integral proof" of the Atiyah-Singer index theorem, see [14][15][16].
In this paper, we use Thm. 1.1 to analyze the short-time asymptotics of the heat kernel at the cut locus. We show that if the set of minimizing geodesics between x and y is a disjoint union of k submanifolds of the space of finite energy paths connecting x and y, having dimensions d 1 , . . . , d k (see Def. 5.1), then under a natural non-degeneracy condition, the heat kernel has an asymptotic expansion of the form as t → 0. In order to derive this result, we show that the convolution product e ν 1 τ * · · · * e ν N τ can be written as an integral over a certain space of piecewise geodesics paths, which can then be evaluated with Laplace's method. See, e.g., [21,26,29], we obtain similar results using methods from stochastic analysis. This paper is organized as follows. First we summarize some facts about the solution theory of the wave equation and introduce the transformation formula, which relates it to the heat equation. Here we also highlight some conditions for the Laplace-type operator that suffice to have the transmutation formula valid on complete manifolds and we use the formula to prove some results on essential self-adjointness. Subsequently, in Sect. 3, we introduce the Hadamard expansion of the solution operator to the heat equation and combine it with the transmutation formula to prove Thm. 1.1. We also briefly demonstrate how the well-known Gaussian estimates from above and below are derived using this technique. In the next section, we give a proof of Theorem. 1.4. In a final section, we reformulate this convolution product as a path integral, which is then analyzed to obtain an asymptotic expansion of the heat kernel p L t (x, y) also in the case that x and y lie in each other's cut locus. In "Appendix," we prove a general version of Laplace's method, which is needed in our considerations.

The wave equation and the transmutation formula
Let M be a complete Riemannian manifold of dimension n and let V be a metric vector bundle over M. A Laplace-type operator L on V is a second-order differential operator acting on sections of V, which in local coordinates is given by where id V is the identity endomorphism on the vector bundle V and (g i j ) is the inverse matrix of the matrix (g i j ) describing the metric in the local coordinates. Any such operator can be written as L = ∇ * ∇ + V for a unique metric connection ∇ and a self-adjoint endomorphism We say that L is formally self-adjoint if it is symmetric on this domain. Given such a Laplace-type operator L, one can consider the wave equation A fundamental feature of the wave equation is the energy estimate, which states that for any compact set K ⊆ M, any m ∈ R and any T > 0, there exists a constant α ∈ R such that for all smooth solutions u of the wave equation with supp u 0 ⊆ K , one has whenever −T ≤ s ≤ t ≤ T (see, e.g., [9,Thm. 8]). From the theory of wave equations follows that there is a family of solution operators G t : D(M, V) −→ D(M, V) such that for ψ ∈ D(M, V), u t := G t ψ solves the wave equation (2.1) with initial conditions u 0 = 0, u 0 = ψ. We also have its derivative G s , which has the property that u t := G t ψ solves the wave equation with initial condition u 0 = ψ, u 0 = 0 (see, e.g., Corollary 14 in [9]).
Instead of the wave equation, we can also consider the heat equation Here we only need to specify an initial condition ψ at time zero to have a unique (bounded) solution. This leads to a solution operator e −t L , mapping the initial condition ψ to the solution u t . The heat equation is related to the wave equation as follows. By the norm bound on G t , the integral on the right-hand (2.5) converges absolutely for each t > 0, and P t is a locally uniformly bounded family of operators. We now verify that P t is a strongly continuous semigroup. First, because γ t integrates to one over the line, we have for all u ∈ L 2 (M, V). Because G s is strongly continuous by assumption and G 0 u = u, the function G s u −u L 2 is continuous in s and vanishes at zero. Now P t u −u L 2 → 0 follows from the well-known fact that γ t → δ 0 as t → 0.
To verify the semigroup property, we use that for any s, t ∈ R and ψ ∈ D(M, V), we have the "trigonometric formula" which can easily be verified by fixing s and noticing that both sides satisfy the wave equation with respect to the variable t and with the same initial conditions. The energy estimate (2.2) implies then that their difference must be zero. Now where the integral over each individual term is absolutely convergent by the bound (2.4) on G u and G u . Because G u is an odd function of u, the term involving G u G v Lψ integrates to zero. Therefore, Hence, P s P t = P t+s on the dense subset D(M, V) ⊂ L 2 (M, V) and by boundedness also on all of L 2 (M, V). This shows that P t is a strongly continuous semigroup of operators.
To see what the infinitesimal generator of P t is, notice that for ψ ∈ D(M, V), the estimates on G t and G t justify the calculation This shows that the infinitesimal generator L (which is always closed for a strongly continuous semigroup) is an extension of the operator L with dom(L) = D(M, V).
In particular, the result is applicable to the compact setting: 2) in this case, since one can take K = M and it is also clear that one can take the same α for each T .
Furthermore, it is well known that on a compact manifold, any Laplace-type operator L has a unique closed extension that is the generator of a strongly continuous semigroup (this follows, e.g., from Lemma 2.16 in [7]).
A consequence of Thm. 2.1 is the following. Proof Let L be a self-adjoint extension of L that generates a strongly continuous semigroup of operators. By the Hille-Yosida theorem, there exists ω ∈ R such that the spectrum of L is contained in [ω, ∞), and e −t L is given in terms of spectral calculus via the absolutely convergent integral where E λ is the spectral measure associated with L. Consider the entire function . Hence, we obtain that By standard properties of the functional calculus, one obtains the estimates g s (L) ≤ e sω , g s (L) ≤ e sω on the operator norms. We now claim that the wave operator G s on D(M, V) is given by G s = g s (L)| D (M,V) . To see this, notice that for any ψ ∈ D(M, V), g s (L)ψ satisfies the wave equation (2.1) with initial conditions g 0 (L)ψ = 0, g 0 (L)ψ = ψ. Hence, u s := G s ψ − g s (L)ψ satisfies the wave equation with initial conditions u 0 = 0, u 0 = 0, which implies u s ≡ 0 by the energy estimate (2.2). The same argument shows that G s = g s (L)| D (M,V) .
By the above, G s and G s satisfy the norm bound (2.4). To see that G s and G s are strongly continuous, we argue as follows: By Lebesgue's theorem of dominated convergence, one obtains that for all Now it is well known that in Hilbert spaces, weak convergence plus convergence of norms implies convergence in norm, so we obtain G s v → G t v in L 2 (M, V). This shows that G s is strongly continuous and the argument for G s is the same. Now by Thm. 2.1, there is some extension L 2 of L with domain containing D(M, V) that generates a strongly continuous semigroup of operators given by the transmutation formula (2.5). However, by Fubini's theorem, where the interior integral is easily found to equal e −tλ , e.g., by expanding g s (λ) into its power series and using standard formulas for the moments of a one-dimensional Gaussian measure (see, e.g., Lemma 2.12 in [7]). Therefore, the semigroup generated by L 2 equals the semigroup generated by L.
These arguments show that self-adjoint extension of L generating a strongly continuous semigroup of operators, this semigroup is given by the transmutation formula (3.1). However, this formula does not depend on the self-adjoint extension (because the operator G s doesn't), so any two strongly continuous operator semigroups generated by self-adjoint extensions of L must coincide. But this implies that also the self-adjoint extensions coincide, because the infinitesimal generator of an operator semigroup is unique.

Example 2.6
For example, if L = ∇ * ∇ + V for some connection ∇ on V and a symmetric endomorphism field V that is bounded from below (meaning that there exists ω ∈ R such that w, V w > ω for all w ∈ V), then L has a self-adjoint extension that generates a strongly continuous semigroup. Namely, because for u ∈ L 2 (M, V), the operator L is semi-bounded and it is well known that it has a self-adjoint extension, called Friedrich extension (see, e.g., [35,VII.2.11]), which satisfies the same bound and therefore generates an operator semigroup by functional calculus. We obtain that in this setting, the Friedrichs extension is the only self-adjoint extension that is the generator of a strongly continuous semigroup.
In particular, this applies to = d * d, the Laplace-Beltrami operator acting on functions.

Example 2.7
The Hodge Laplacian L = (d + d * ) 2 on differential forms is a positive operator and hence has a self-adjoint extension generating a strongly continuous semigroup by the same argument. By Thm. 2.5, this is the only self-adjoint extension generating a strongly continuous semigroup of operators. In fact, it is the only self-adjoint extension, by Thm. 2.4 in [33]. For the same reason, for any self-adjoint Dirac-type operator D, the corresponding Laplacian D 2 has a unique self-adjoint extension generating a strongly continuous semigroup of operators. Also in this case, it is known that D and D 2 are even essentially self-adjoint (i.e., they have unique self-adjoint extensions), compare [36].

Example 2.8
In contrast, there are formally self-adjoint Laplace-type operators that do not admit any self-adjoint extension. For example, the operator L = − − x 4 on M = R does not admit a self-adjoint extension (see Ex. 3 on p. 86 in [8]). There are also essentially self-adjoint Laplace-type operators which do not generate a strongly continuous family of operators, see, e.g., [32].

Remark 2.9
Our observations show that matters can be quite subtle on general complete manifolds: A formally self-adjoint Laplace-type operator need not have a self-adjoint extension, nor need it be unique. Furthermore, not all self-adjoint extensions generate a strongly continuous semigroup of operators. (They do if and only if the spectrum is bounded from below.) However, there is at most one self-adjoint extension that generates a strongly continuous semigroup of operators. We do not know of an example of a formally self-adjoint Laplace-type operator that admits two different self-adjoint extensions, one of which generates a strongly continuous semigroup and the other doesn't. (By Thm. 2.5, not both of them can generate a strongly continuous semigroup of operators.)

Heat kernel asymptotics
In this section, we prove the following more general version of Thm. 1.1.
The proof will use the transmutation formula (3.1), which in terms of integral kernels translates into where G (s, x, y) is the Schwartz kernel of the operator G s . This integral is meant as a distributional integral, i.e., for test functions ϕ ∈ D(M, V * ), ψ ∈ D(M, V), we set Let us verify that this indeed defines a distribution on M × M for each t > 0, provided that the assumptions of Thm. 2.1 hold. Namely, by the estimate on G s , we have which shows that the integral (3.1) is absolutely convergent. We furthermore have The wave kernel G(t, x, y) has an asymptotic expansion, the Hadamard expansion, which describes its singularity structure. To state the result, we introduce the Riesz distributions R(α; t, x, y) ∈ D(M M). Namely, for Re(α) > n + 1, we set j (x, y)R(2 + 2 j; t, x, y), (3.4) where the j (x, y) ∈ C ∞ (M M, V V * ) are coefficients determined by certain transport equations. The asymptotic expansion (3.4) is meant in the sense that the difference can be made arbitrarily smooth by increasing the number ν of correction terms; in fact, Furthermore, the fact that the wave equation has finite propagation speed (i.e., G(t, x, y) ≡ 0 on the region where |t| < d(x, y)) implies that when ν is so large that δ ν is C k , one has the estimate  where e t (x, y) is the Euclidean heat kernel, defined in (1.2). In particular, the distributional integral on the left-hand side actually yields a smooth function.
Proof For Re(α) > n + 1, consider the absolutely convergent integral Performing the substitution u 2 = s 2 − d(x, y) 2 which transforms the interval (d(x, y), ∞) into the interval (0, ∞), we have sds = udu. Therefore, we obtain Put together, we arrive at (3.8) Until now, we have restricted ourselves to the case Re α > n + 1. However, for both sides of the last equation, if we pair them with a test function ϕ ∈ D(M M), the result will be an entire holomorphic function in α. Because they coincide for Re α > n + 1, they must coincide everywhere, by the identity theorem for holomorphic functions.
The statement of the lemma is the particular result for α = 2 + 2 j, j ∈ N 0 .
Proof (of Thm. 3.1) Integrating by parts in (3.1), which is justified by the estimate (2.4), we obtain where the identity is to be interpreted in the distributional sense. Now for any ν ∈ N, we have where δ ν (t, x, y) is in C k whenever ν ≥ (n +1)/2+k. By Lemma 3.2, the first term evaluates to It remains to estimate the error term. Because G t = −G −t and the Riesz distributions are odd in t, the remainder term δ ν (t, x, y) is an odd function in the t variable. We conclude as δ ν (s, x, y) = 0 if s < d(x, y), because of (3.6). Substituting s = u 2 + d(x, y) 2 as before, one obtains Settingδ ν (u, x, y) := δ ν ( u 2 + d(x, y) 2 , x, y) one has thatδ ν is C k whenever δ ν is C k , and from (3.6) follows the estimate if ν is large enough, depending on l and m. The estimate (3.10) shows that these manipulations make sense when ν is large enough, i.e., in this case, the integral is absolutely convergent and uniformly bounded independent of t. Therefore, for any ν, one can findν ≥ ν large enough so that where the estimate is uniform for (x, y) in compact subsets of M M and t ≤ T . However, the calculation shows that in factν = ν suffices.

Corollary 3.3 The heat kernel coefficients satisfy the symmetry relation
where * j are the heat kernel coefficients for the heat kernel p * t of the formally adjoint operator L * and ( * j (y, x)) * denotes the fiberwise metric adjoint of * j (y, x).
Proof By Theorem 1.1, this follows from the fact that the heat kernel itself satisfies the same symmetry relation by Prop. 2.17 (2) in [7]. Note that this argument does not work if one only knows (1.4).

Remark 3.4 Corollary 3.3 is not at all obvious from the defining transport equations for the
j . The result was previously proved in the scalar case by Moretti [27,28] for the heat equation and the Hadamard coefficients by approximating the given metric by real analytic metrics. However, for the heat kernel coefficients, this comes out directly from Thm. 1.1.
In the remainder of this section, we demonstrate how to obtain Gaussian estimates on p L t using our techniques.

Theorem 3.5 (Gaussian upper bound) Let L be a formally self-adjoint Laplace-type operator acting on sections of a vector bundle V over a compact manifold M and let p t be its heat kernel. Then for any T > 0, there exists a constant C > 0 such that
for all x, y ∈ M, whenever t ≤ T .

Remark 3.6
In fact, the upper bound can be improved to have a pre-factor of t −n+1/2 instead of t −n−1 in the case j = m = l = 0 [20,Thm. 5.3.4]. This result is then sharp, as seen, e.g., by the example of two antipodal points of a sphere [20,Example 5.3.3]. Of course, if (x, y) ∈ M M, then the correct exponent is t −n/2 near (x, y), by Thm. 1.1. G (t, x, y) of G t as a distribution on R × M × M has order at most n + 1, i.e., it is a sum of (n + 1)-st distributional derivatives of continuous functions. This follows from the asymptotic expansion (3.4) and the fact that the Riesz distribution R(2 + 2 j) has distributional order at most n + 1 − 2 j for 0 ≤ 2 j < n + 1 and is continuous if 2 j ≥ n + 1 [6, Section 1.2]. The wavefront set of G (t, x, y) is contained in the characteristics of the wave operator ∂ tt + L, (i.e., the "light cone") which are transversal to the submanifolds R × {(x, y)} ⊂ R × M × M. Therefore, one can restrict G to these submanifolds, so that for (x, y) ∈ M × M fixed, G (t, x, y) is a distribution on R of order at most n + 1 in the variable t. x, y) is a distribution of order at most k := n + 1 + m + l + j on R. This means that

Proof (sketch) The Schwartz kernel
x, y) (3.11) in the sense of distributions for some L 1 function f (s, x, y). Using the transmutation formula (3.1) and integration by parts, we obtain which gives a pre-factor of order −k in t. Here, the integration by parts is justified by standard energy estimates. Differentiating j times by t gives another pre-factor of order −2 j in t.
The result now follows from the fact that G (s, x, y) and hence also f (s, x, y) is equal to zero for |s| < d(x, y), by finite propagation speed of the wave equation.

Remark 3.7
If n is even, R(2 + 2 j) is in fact of order n − 2 j. This improves the estimate of Thm. 3.5 to a pre-factor of t −(n+2 j+m+l) on the right-hand side in even dimensions.
On the other hand, by Lemma 5.7 2 , the convolution product e ν t/N * · · · * e ν t/N can be written as an integral over the manifold H xy;τ (M) of piecewise geodesics (introduced in Sect. 5), where E is the energy functional (5.1) and ϒ τ,ν (t, γ ) is some smooth function on H xy;τ (M), depending polynomially on t. An investigation of the integral using Laplace's method (see "Appendix 1") shows that for N large, where ε > 0 is independent of x and y, where one uses that ϒ τ,ν (0, γ ) > 0 for all minimal geodesics γ connecting x and y, if N is large enough.

Convolution approximation
In this section, we prove Thm. 1.4. Throughout, M is a compact Riemannian manifold and L is a Laplace-type operator, acting on sections of a metric vector bundle V over M.
The proof relies the following lemma.
whenever 0 ≤ s 0 < s 1 ≤ t ≤ T and s 1 − s 0 ≤ tδ. Here p t denotes the heat kernel of the Laplace-Beltrami operator on M.

Proof Set
and put By Thms. 3.5 and 3.8, there exist constants C 1 , C 2 > 0 such that for all 0 < t ≤ T and all x, y ∈ M, we have (4.1) Using this, we obtain Now set for any ε with 0 < ε < ε Then on the set where ϕ(d(z 0 , z 1 )) = 0, i.e., d(z 0 , z 1 ) ≥ R, we have whenever s 1 − s 0 ≤ tδ the estimate Hence, under this restriction on s 1 − s 0 and using that the function p t (x, −) integrates to one for each x ∈ M, as well as (4.1), we have for each 0 < t ≤ T that if the constants C 3 , C 4 are chosen appropriately.

Remark 4.2
The proof above shows that one can choose δ as in Thm. 1.4 in order that the statement of Lemma 4.1 holds.
We can now prove Thm. 1.4.
Proof (of Thm. 1.4) Throughout the proof, write j := j τ for abbreviation. By the Markov property of the heat kernel, we have p L t = p L s * p L t−s for all 0 < s < t. We obtain that since the sum on the right-hand side telescopes. By the Hess-Schrader-Uhlenbrock estimate [19], we have | p L t | ≤ e αt p t for some constant α ∈ R, where p t denotes the heat kernel of the Laplace-Beltrami operator (here we use self-adjointness of the operator L). Similarly, t p t (x, y) for some α > 0, where we used Thm. 1.1 and the Gaussian estimate from below, Thm. 3.8. Here, χ is a smooth function with χ(r ) = 1 near zero and support contained in [0, inj(M)), with inj(M) the injectivity radius. Therefore, Now, by Thm. 1.1 and the Gaussian estimate from below, Therefore, where R is such that χ(r ) = 1 for 0 ≤ r ≤ R. The first term can be estimated by By Lemma 4.1, whenever |τ | ≤ δt, the second term can be estimated by with , > 0. This finishes the proof.

Heat kernel asymptotics at the cut locus
In this section, we use the convolution approximation from Thm. 1.4 to obtain short-time asymptotic expansions of the heat kernel also in the case that x, y ∈ M lie in each other's cut locus. As we will see, the form of such an asymptotic expansion depends on the behavior of the energy functional near its critical points on the space of paths between x and y.

Remark 5.3
In particular, if (x, y) ∈ M M so that min xy = {γ } with γ the unique minimizing geodesic between x and y, then we recover the asymptotic expansion from before, Thm. 1.1.

Remark 5.4
The Hessian of the energy at an element γ ∈ min xy can be explicitly calculated and is closely related to the Jacobi equation, see, e.g., [25,Section 13].
Remark 5.5 Thm. 5.2 can be generalized to the case that min xy is a degenerate submanifold of H xy (M). In this case, the explicit form of the asymptotic expansion depends on the type of degeneracy of E. In general, it can become quite complicated; for example it may contain logarithmic terms. For a discussion of this, see [26, pp. 20-24]. is a diffeomorphism between the two. For our purpose, it doesn't matter which Riemannian metric (or volume) we put on H xy;τ (M); for simplicity we take the one that makes ev τ an isometry.

Lemma 5.7
The heat convolution product from Thm. 1.4 can be written as an integral over H xy;τ (M). More specifically, for a partition τ = {0 = τ 0 < τ 1 < · · · < τ N = t}, denote bỹ τ the corresponding partition of the interval [0, 1], given byτ j = τ j /t. Then we have where the integrand ϒτ ,ν (t, γ ) is a certain smooth and compactly supported function on H xy;τ (M) with values in Hom(V y , V x ) that depends polynomially on t.
Proof Notice that for the path γ ∈ H xy;τ (M) with γ (τ j ) = x j , we have Now because the approximate heat kernel e ν t is supported in M M (by choice of the cutoff function present in its definition), the convolution e ν 1 τ * · · · * e ν N τ can be written as an integral over M (N −1) , e ν 1 τ * · · · * e ν N τ (x, y) We obtain formula (5.2), where This finishes the proof.
The explicit formula for ϒτ ,ν is entirely unimportant for our purposes; we only take from it that ϒτ ,ν is a smooth, compactly supported function on H xy;τ (M) that depends polynomially on t.
Below, we will always write τ instead ofτ for a partition of the interval [0, 1].
Because ν was arbitrary, we obtain that p L t (x, y)/e t (x, y) has a complete asymptotic expansion of the claimed form, with the coefficients j,l (x, y) given by the formula (5.5) for ν large enough and |τ | small enough. a). Laplace's method is simpler in the sense that here, only critical points which are minima contribute to the asymptotic expansion, while for integrals with imaginary exponent, all critical points contribute. Compare, e.g., [1] or [ Notice that ε > 0 because A ∩ = ∅. Therefore, if we choose 0 < ε < ε and C > 0 appropriately.
Proof (of Thm. A.1) We may write the integral over as a sum of integrals over open subsets 1 , . . . , k such that the union of the l is dense in , and such that l ⊂ l for each l = 1, . . . , k. The asymptotic expansion of the integral over will then the be sum of the asymptotic expansions of the integrals over the manifolds l . Therefore, we may assume that k = 1, i.e., is a non-degenerate submanifold of dimension d.
Let N := dim( ) and let A as in (A.3). Since A is compact, we may without loss of generality assume that also and hence is compact.
This can be proved using the implicit function theorem, compare, e.g., Lemma 1.2.2 in [13]. Clearly, we have dκ| (x,0) = id x . Furthermore, we may assume that A ⊂ U . Namely otherwise, we can choose a cutoff function χ ∈ C ∞ c (U ) that is equal to one on a neighborhood of and split I (t, a) = I (t, χa) + I (t, (1 − χ)a), where the second summand does not contribute to the asymptotic expansion because of Lemma A.3.
We now may use the transformation formula to obtain Now on the vector spaces N x , define the Q-Laplacian Q by the formula This patches together to a smooth differential operator on N satisfying Therefore, integrating by parts, we obtain where for f ∈ C ∞ (U ), we set so that P is some second-order differential operator. Let J (t, a) := (4πt) d/2 I (t, a). Then by Taylor's formula and the Leibnitz rule, for all ε > 0 and ν ∈ N, where R ν (ε, t) = Because of (A.5), we may take the limit ε → 0 to obtain lim ε→0 J ε, P j−i a (i) = P j−i a (i) (0, x) det ∇ 2 φ| N x 1/2 dx. Therefore, for any ν ∈ N 0 , where the remainder term is of order t ν+1 . This finishes the proof.