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

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## Abstract

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.

## Keywords

Heat kernel Laplace operator Manifold Asymptotic expansion Heat equation Wave equation Transmutation formula## 1 Introduction and main results

*M*be a compact Riemannian manifold of dimension

*n*and let

*L*be a Laplace-type operator, acting on sections of a vector bundle \(\mathcal {V}\) over

*M*. For \(t>0\), the heat kernel \(p_t^L\) of

*L*is a smooth section of the bundle \(\mathcal {V}\boxtimes \mathcal {V}^*\) over \(M \times M\) (the vector bundle with fiber \(\mathrm {Hom}(V_y, V_x)\) over the point \((x, y) \in M\times M\)). It is well known that for \(x, y \in M\) close, the heat kernel has an asymptotic expansion of the form

*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

*L*be a Laplace-type operator, acting on sections of a vector bundle \(\mathcal {V}\) over a compact Riemannian manifold

*M*. Then for any compact subset

*K*of \(M \bowtie M\), any \(T>0\) and any numbers \(\nu , k, l, m \in \mathbb {N}_0\), there exists a constant \(C>0\) such that

In the theorem, \(\nabla _x\) and \(\nabla _y\) denote the covariant derivative with respect to the *x* (respectively *y*) variable, where we use any metric connection on the bundle \(\mathcal {V}\) (changing the connection only alters the constant *C* on the right-hand side).

### Corollary 1.2

*x*,

*y*) over compact subsets of \(M \bowtie M\). 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 \(C t^{\nu +1} e_t(x, y)\), 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 \(\Phi _j\) and was previously proved using substantially more involved arguments (see Remark 3.4).

### Theorem 1.3

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.

*approximate heat kernels*\(e_t^\nu (x, y)\) by

*M*). If for general smooth kernels \(k, \ell \in C^\infty (M \times M, \mathcal {V}\boxtimes \mathcal {V}^*)\), we define their convolution \(k * \ell \) by

### Theorem 1.4

*L*be a formally self-adjoint Laplace-type operator, acting on sections of a metric vector bundle \(\mathcal {V}\) over a compact Riemannian manifold

*M*. Then for any \(\delta > 0\) with

*t*] with \(|\tau | \le \delta t\), where \(p_t^\Delta \) is the heat kernel of the Laplace–Beltrami operator on

*M*. Here we used the notations \(\Delta _j\tau := \tau _j - \tau _{j-1}\) and \(|\tau | := \max _{1 \le j \le N} \Delta _j \tau \) for the increment, respectively the mesh of a partition \(\tau \).

This approximation result can be used in different regimes: If one fixes \(t>0\), one can make the \(C^k\) difference in between \(p_t^L\) and \(e_{\Delta _1\tau }^\nu * \cdots * e_{\Delta _N \tau }^\nu \) smaller than any given \(\varepsilon >0\), by choosing a partition \(\tau \) fine enough. On the other hand, by choosing \(\nu \) large enough, this error can be made uniform in *t*.

*a posteriori*estimate, in the sense that the error depends on \(p_t^\Delta (x, y)\), 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], \(p_t^\Delta (x, y) \le C t^{-n/2+1/2}e_t(x, y)\), which holds on compact Riemannian manifolds: One gets that one can replace the result of Thm. 1.4 by the estimate

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].

*x*and

*y*is a disjoint union of

*k*submanifolds of the space of finite energy paths connecting

*x*and

*y*, having dimensions \(d_1, \ldots , d_k\) (see Def. 5.1), then under a natural non-degeneracy condition, the heat kernel has an asymptotic expansion of the form

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 \(p_t^L(x, y)\) 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.

## 2 The wave equation and the transmutation formula

*M*be a complete Riemannian manifold of dimension

*n*and let \(\mathcal {V}\) be a metric vector bundle over

*M*. A Laplace-type operator

*L*on \(\mathcal {V}\) is a second-order differential operator acting on sections of \(\mathcal {V}\), which in local coordinates is given by

*V*[7, Section 2.1]. For example, if

*L*is acting on functions (i.e., sections of the trivial line bundle), we could have \(L = \Delta + v\) with \(\Delta \) the Laplace–Beltrami operator and

*v*some potential function. Considered as an unbounded operator on \(L^2(M, \mathcal {V})\), a natural domain for

*L*is the space \({\mathscr {D}}(M, \mathcal {V}) := C^\infty _c(M, \mathcal {V})\), the space of smooth, compactly supported sections of the bundle \(\mathcal {V}\) (which, when necessary, is endowed with the usual test function topology). We say that

*L*is formally self-adjoint if it is symmetric on this domain.

*L*, one can consider the

*wave equation*

*energy estimate*, which states that for any compact set \(K \subseteq M\), any \(m \in \mathbb {R}\) and any \(T>0\), there exists a constant \(\alpha \in \mathbb {R}\) such that for all smooth solutions

*u*of the wave equation with \(\mathrm {supp}\, u_0 \subseteq K\), one has

From the theory of wave equations follows that there is a family of solution operators \(G_t: {\mathscr {D}}(M, \mathcal {V}) \longrightarrow {\mathscr {D}}(M, \mathcal {V})\) such that for \(\psi \in {\mathscr {D}}(M, \mathcal {V})\), \(u_t := G_t \psi \) solves the wave equation (2.1) with initial conditions \(u_0 = 0\), \(u_0^\prime = \psi \). We also have its derivative \(G_s^\prime \), which has the property that \(u_t := G_t^\prime \psi \) solves the wave equation with initial condition \(u_0 = \psi \), \(u^\prime _0 = 0\) (see, e.g., Corollary 14 in [9]).

*heat equation*

### Theorem 2.1

*M*be a complete Riemannian manifold and let

*L*be a Laplace-type operator, acting on sections of a metric vector bundle \(\mathcal {V}\) over

*M*. Suppose that the wave operators \(G_t\) and \(G_t^\prime \) defined on \({\mathscr {D}}(M, \mathcal {V})\) extend to strongly continuous families of operators on \(L^2(M, \mathcal {V})\) satisfying the norm bound

*L*with \({\mathrm {dom}}(L)={\mathscr {D}}(M, \mathcal {V})\).

### Remark 2.2

Of course, the continuous extensions of \(G_t\) respectively \(G_t^\prime \), if they exist, are unique, since \({\mathscr {D}}(M, \mathcal {V})\) is dense in \(L^2(M, \mathcal {V})\).

### Remark 2.3

The same result is true when \(L^2(M, \mathcal {V})\) is replaced by any Banach space *E* of distributions containing \({\mathscr {D}}(M, \mathcal {V})\) as a dense subset and such that the inclusion of *E* into \({\mathscr {D}}^\prime (M, \mathcal {V})\) is continuous.

### Proof

*s*and vanishes at zero. Now \( \Vert P_t u - u\Vert _{L^2}\rightarrow 0\) follows from the well-known fact that \(\gamma _t \rightarrow \delta _0\) as \(t\rightarrow 0\).

*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

*u*, the term involving \(G_u G_v L\psi \) integrates to zero. Therefore,

*L*with \({\mathrm {dom}}(L) = {\mathscr {D}}(M, \mathcal {V})\). \(\square \)

In particular, the result is applicable to the compact setting:

### Lemma 2.4

For any Laplace-type operator acting on sections of a metric vector bundle over a compact Riemannian manifold, the assumptions of Thm. 2.1 are satisfied.

### Proof

The bound (2.4) follows directly from the energy estimate (2.2) in this case, since one can take \(K = M\) and it is also clear that one can take the same \(\alpha \) for each *T*. \(\square \)

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.

### Theorem 2.5

Let *L* be a formally self-adjoint Laplace-type operator, acting on sections of a metric vector bundle over a complete Riemannian manifold. Considered as an unbounded symmetric operator with domain \({\mathscr {D}}(M, \mathcal {V})\), *L* admits at most one self-adjoint extension \(\overline{L}\) that generates a strongly continuous semigroup of operators. If there is such an extension, then the assumptions of Thm. 2.1 are satisfied.

### Proof

*L*that generates a strongly continuous semigroup of operators. By the Hille–Yosida theorem, there exists \(\omega \in \mathbb {R}\) such that the spectrum of \(\overline{L}\) is contained in \([\omega , \infty )\), and \(\hbox {e}^{-t\overline{L}}\) is given in terms of spectral calculus via the absolutely convergent integral

We now claim that the wave operator \(G_s\) on \({\mathscr {D}}(M, \mathcal {V})\) is given by \(G_s = g_s(\overline{L})|_{{\mathscr {D}}(M, \mathcal {V})}\). To see this, notice that for any \(\psi \in {\mathscr {D}}(M, \mathcal {V})\), \(g_s(\overline{L}) \psi \) satisfies the wave equation (2.1) with initial conditions \(g_0(\overline{L})\psi = 0\), \(g_0^\prime (\overline{L}) \psi = \psi \). Hence, \(u_s := G_s\psi - g_s(\overline{L}) \psi \) satisfies the wave equation with initial conditions \(u_0 = 0\), \(u_0^\prime = 0\), which implies \(u_s \equiv 0\) by the energy estimate (2.2). The same argument shows that \(G_s^\prime = g_s^\prime (\overline{L})|_{{\mathscr {D}}(M, \mathcal {V})}\).

By the above, \(G_s\) and \(G_s^\prime \) satisfy the norm bound (2.4). To see that \(G_s\) and \(G_s^\prime \) are strongly continuous, we argue as follows: By Lebesgue’s theorem of dominated convergence, one obtains that for all \(u, v \in L^2(M, \mathcal {V})\), one has \((u, G_s v)_{L^2} \rightarrow (u, G_t v)_{L^2}\) as \(s \rightarrow t\), i.e., \(G_sv \rightarrow G_t v\) weakly. Similarly, \(\Vert G_s v\Vert _{L^2} = (v, G_s^2 v)_{L^2} \longrightarrow (v, G_t^2 v)_{L^2} = \Vert G_t v\Vert _{L^2}\). Now it is well known that in Hilbert spaces, weak convergence plus convergence *of* norms implies convergence *in* norm, so we obtain \(G_s v \rightarrow G_t v\) in \(L^2(M, \mathcal {V})\). This shows that \(G_s\) is strongly continuous and the argument for \(G_s^\prime \) is the same.

*L*with domain containing \({\mathscr {D}}(M, \mathcal {V})\) that generates a strongly continuous semigroup of operators given by the transmutation formula (2.5). However, by Fubini’s theorem,

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. \(\square \)

### Example 2.6

*V*that is bounded from below (meaning that there exists \(\omega \in \mathbb {R}\) such that \(\left\langle w, Vw\right\rangle > \omega \) for all \(w \in \mathcal {V}\)), then

*L*has a self-adjoint extension that generates a strongly continuous semigroup. Namely, because for \(u \in L^2(M, \mathcal {V})\),

*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 \(\Delta = d^* d\), 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 \(L = -\Delta - x^4\) on \(M = \mathbb {R}\) 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.)

## 3 Heat kernel asymptotics

In this section, we prove the following more general version of Thm. 1.1.

### Theorem 3.1

*L*be a Laplace-type operator, acting on sections of a vector bundle \(\mathcal {V}\) over a complete Riemannian manifold

*M*. Suppose that the assumptions of Thm. 2.1 are satisfied (e.g., when

*M*is compact or

*L*is formally self-adjoint and semi-bounded). Then for any compact subset

*K*of \(M \bowtie M\), any \(T>0\) and any numbers \(\nu , k, l, m \in \mathbb {N}_0\), there exists a constant \(C>0\) such that

*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*\(R(\alpha ; t, x, y) \in {\mathscr {D}}(M \bowtie M)\). Namely, for \(\mathrm {Re}(\alpha )> n+1\), we set

*G*(

*t*,

*x*,

*y*) has the asymptotic expansion [6, Ch. 2]

### Lemma 3.2

### Proof

The statement of the lemma is the particular result for \(\alpha = 2 + 2j\), \(j \in \mathbb {N}_0\). \(\square \)

### Proof

*t*, the remainder term \(\delta ^\nu (t, x, y)\) is an odd function in the

*t*variable. We conclude

*x*,

*y*in compact subsets of \(M \bowtie M\) and \(u\le T\). Now the function \(\hbox {e}^{-u^2/4t}\) satisfies

*l*and

*m*. The estimate (3.10) shows that these manipulations make sense when \(\nu \) is large enough, i.e., in this case, the integral is absolutely convergent and uniformly bounded independent of

*t*. Therefore, for any \(\nu \), one can find \(\tilde{\nu } \ge \nu \) large enough so that

*x*,

*y*) in compact subsets of \(M \bowtie M\) and \(t \le T\). However, the calculation

### Corollary 3.3

### 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). \(\square \)

### Remark 3.4

Corollary 3.3 is not at all obvious from the defining transport equations for the \(\Phi _j\). 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 \(p_t^L\) using our techniques.

### Theorem 3.5

*L*be a formally self-adjoint Laplace-type operator acting on sections of a vector bundle \(\mathcal {V}\) 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

### 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 \((x, y) \in M \bowtie M\), then the correct exponent is \(t^{-n/2}\) near (*x*, *y*), by Thm. 1.1.

### Proof

*t*. Similarly, \(\frac{\partial ^j}{\partial t^j}\nabla _x^k \nabla _y^l G^\prime (t, x, y)\) is a distribution of order at most \(k:=n+1+m+l+j\) on \(\mathbb {R}\). This means that

*f*(

*s*,

*x*,

*y*). Using the transmutation formula (3.1) and integration by parts, we obtain

*t*. Here, the integration by parts is justified by standard energy estimates. Differentiating

*j*times by

*t*gives another pre-factor of order \(-2j\) in

*t*.

The result now follows from the fact that \(G^\prime (s, x, y)\) and hence also *f*(*s*, *x*, *y*) is equal to zero for \(|s| < d(x, y)\), by finite propagation speed of the wave equation. \(\square \)

### Remark 3.7

If *n* is even, \(R(2+2j)\) is in fact of order \(n-2j\). This improves the estimate of Thm. 3.5 to a pre-factor of \(t^{-(n+2j+m+l)}\) on the right-hand side in even dimensions.

### Theorem 3.8

*M*be a compact Riemannian manifold and let

*L*be a scalar Laplace-type operator

*L*, i.e., a Laplace-type operator acting on functions on

*M*. Then for any \(T>0\), there exists a constant \(C>0\) such that

### Proof

*L*defined in (1.5). Because of Thm. 1.1, there exists \(C>0\) such that for any \(N \in \mathbb {N}\), we have

^{2}, the convolution product \(e_{t/N}^\nu * \cdots * e_{t/N}^\nu \) can be written as an integral over the manifold \(H_{xy;\tau }(M)\) of piecewise geodesics (introduced in Sect. 5), where \(\tau := \{ 0< \frac{1}{N}< \frac{2}{N}< \cdots< \frac{N-1}{N} < 1 \}\) denotes the equidistant partition of the interval [0, 1]. More specifically,

*E*is the energy functional (5.1) and \(\Upsilon _{\tau , \nu }(t, \gamma )\) is some smooth function on \(H_{xy;\tau }(M)\), depending polynomially on

*t*. An investigation of the integral using Laplace’s method (see “Appendix 1”) shows that for

*N*large,

*x*and

*y*, where one uses that \(\Upsilon _{\tau , \nu }(0, \gamma )>0\) for all minimal geodesics \(\gamma \) connecting

*x*and

*y*, if

*N*is large enough.\(\square \)

### Remark 3.9

There is a rich literature containing Gaussian bounds for the Laplace–Beltrami operator. In the stochastic literature, two-sided estimates can be found, e.g., in [26, 20, Thm. 5.3.4] and [4]. Using analytic methods, the Gaussian estimate from above is derived, e.g., in [12, 18, Thm. 15.14] and [11].

## 4 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 \(\mathcal {V}\) over *M*.

The proof relies the following lemma.

### Lemma 4.1

*M*.

### Proof

### Remark 4.2

The proof above shows that one can choose \(\delta \) as in Thm. 1.4 in order that the statement of Lemma 4.1 holds.

We can now prove Thm. 1.4.

### Proof

*L*). Similarly,

*R*is such that \(\chi (r) = 1\) for \(0 \le r \le R\). The first term can be estimated by

## 5 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 \(x, y \in M\) 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*.

*energy functional*

Let \(\Gamma _{xy}^{\min } \subset H_{xy}(M)\) denote the set of length minimizing geodesics between the points \(x, y \in M\). It is well known that for each \(\gamma \in \Gamma _{xy}^{\min }\), we have \(E(\gamma ) = d(x, y)^2/2\), and conversely, the set \(\Gamma _{xy}^{\min }\) is exactly the set of global minima of *E* on \(H_{xy}(M)\). Moreover, \(\Gamma _{xy}^{\min }\) is compact in \(H_{xy}(M)\) [23, Prop. 2.4.11].

### Definition 5.1

Let \(x, y \in M\). We say that \(\Gamma _{xy}^{\min }\) is a *non-degenerate submanifold*, if it is a submanifold of \(H_{xy}(M)\), and if furthermore for each \(\gamma \in \Gamma _{xy}^{\min }\), the Hessian of *E* is non-degenerate when restricted to a complementary subspace to the tangent space \(T_\gamma \Gamma _{xy}^{\min }\).

This is just the well-known Morse–Bott condition on the energy function near the submanifold \(\Gamma _{xy}^{\min }\).

### Theorem 5.2

*M*be a compact manifold and let

*L*be a self-adjoint Laplace-type operator, acting on sections of a metric vector bundle \(\mathcal {V}\) over

*M*. For \(x, y \in M\), assume that the set \(\Gamma _{xy}^{\min }\) is a disjoint union of

*k*non-degenerate submanifolds of dimensions \(d_1, \ldots , d_k\). Then the heat kernel has the complete asymptotic expansion

### Remark 5.3

In particular, if \((x, y) \in M \bowtie M\) so that \(\Gamma _{xy}^{\min } = \{\gamma \}\) with \(\gamma \) 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 \(\gamma \in \Gamma _{xy}^{\min }\) 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 \(\Gamma _{xy}^{\min }\) 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].

### Example 5.6

A prototypical example where \(\Gamma _{xy}^{\min }\) is a non-degenerate submanifold of dimension greater than zero is when *x* and *y* are antipodal points on a sphere. In this case, \(\dim \Gamma _{xy}^{\min } = n-1\). For an explicit calculation of \(\Phi _0(x, y)\) in this case, see [20, Example 5.3.3].

### Lemma 5.7

*t*.

### Proof

The explicit formula for \(\Upsilon ^{\tilde{\tau }, \nu }\) is entirely unimportant for our purposes; we only take from it that \(\Upsilon ^{\tilde{\tau }, \nu }\) is a smooth, compactly supported function on \(H_{xy;\tau }(M)\) that depends polynomially on *t*.

Below, we will always write \(\tau \) instead of \(\tilde{\tau }\) for a partition of the interval [0, 1].

### Proof (of Thm. 5.2)

*P*on \(H_{xy;\tau }(M)\). Here, \(\det (\nabla ^2 E|_{N_\gamma \Gamma _l})^{1/2}\) denotes the determinant of \(\nabla ^2 E|_\gamma \), restricted to the normal space \(N_\gamma \Gamma _l\) of \(T_\gamma \Gamma _l\) in \(T_\gamma H_{xy;\tau }(M)\). In particular, if we set \(d := \max _{1\le l \le k} d_l\), there exists a constant \(C_0>0\) such that

*M*, some \(\nu \ge n/2 - k/2 -1\) and \(|\tau |\le \delta \), we get

Because \(\nu \) was arbitrary, we obtain that \(p_t^L(x, y)/e_t(x, y)\) has a complete asymptotic expansion of the claimed form, with the coefficients \(\Phi _{j, l}(x, y)\) given by the formula (5.5) for \(\nu \) large enough and \(|\tau |\) small enough. \(\square \)

## Footnotes

## Notes

### Acknowledgements

Open access funding provided by Max Planck Society. I would like to thank Christian Bär, Rafe Mazzeo, Franziska Beitz, Florian Hanisch and Ahmad Afuni for helpful discussions. Furthermore, I am indebted to Potsdam Graduate School, The Fulbright Program, SFB 647 and the Max-Planck-Institute for Mathematics in Bonn for financial support.

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