Heat Flow Out of a Compact Manifold

Abstract

We discuss the heat content asymptotics associated with the heat flow out of a smooth compact manifold in a larger compact Riemannian manifold. Although there are no boundary conditions, the corresponding heat content asymptotics involve terms localized on the boundary. The classical pseudo-differential calculus is used to establish the existence of the complete asymptotic series, and methods of invariance theory are used to determine the first few terms in the asymptotic series in terms of geometric data. The operator driving the heat process is assumed to be an operator of Laplace type.

1 Introduction

1.1 Basic Assumptions

We adopt the following notational conventions. We let $(M,g)$ be a smooth Riemannian manifold of dimension $m$. Let $\Omega \subset M$ be a smooth compact submanifold of $M$ which has dimension $m$ as well. We suppose that the boundary of $\Omega$ is non-empty and smooth. In this paper, we shall investigate the heat flow out of $\Omega$ into $M\setminus \Omega$ where $(M,g)$ satisfies exactly one of the following three conditions:

• (1) $M$ is a compact and without boundary.

• (2) $(M,g)=(\mathbb {R}^m,g_e)$ where $g_e$ is the usual Euclidean inner metric on $\mathbb {R}^m$.

• (3) $M$ is a compact submanifold of $\mathbb {R}^m$ with smooth boundary and $g=g_e|_M$.

The case of flat space is fundamental in this subject and hence we have concentrated on that setting in (2) and (3) to avoid technical difficulties. However, the results of this paper hold more generally.

1.2 Heat Content

Let $\Delta _g$ be the associated Laplace–Beltrami operator acting on smooth functions on $M$. The heat equation on $M$ takes the form

$$\begin{aligned} \Delta _gu+\frac{\partial u}{\partial t}=0,\ x\in M,\ t>0, \end{aligned}$$

(1.1)

with initial condition

$$\begin{aligned} u(x;0)=\phi (x),\ x\in M, \end{aligned}$$

(1.2)

where $\phi :M\rightarrow \mathbb R$ is continuous. Here $u(x;t)$ represents the temperature at a point $x\in M$ at time $t$ if $M$ has initial temperature profile $\phi$. We say that $K(x,\tilde{x};t)$ is a fundamental solution if we have:

$$\begin{aligned} u(x;t)=\int \limits _MK(x,\tilde{x};t)\phi (\tilde{x})d\tilde{x}. \end{aligned}$$

We say that $(M,g)$ is stochastically complete if

$$\begin{aligned} \int \limits _MK(x,\tilde{x};t)d\tilde{x}=1\text { for all }x\in M,\ \ t>0. \end{aligned}$$

In order to guarantee that Eqs. (1.1) and (1.2) have a unique solution, we have to impose some conditions on the geometry of $M$. These are summarized in the following result (see Chapter VII of [5] and the references therein):

Theorem 1.1

• (1) If $(M,g)$ is compact without boundary or if $(M,g)$ is complete and with Ricci curvature bounded from below, then there exists a unique minimal positive fundamental solution $K$. Moreover, $(M,g)$ is stochastically complete.

• (2) If $(M,g)$ is compact with non-empty smooth boundary $\partial M$, then there exists a unique minimal positive fundamental solution $K$. In this case $K$ is the Dirichlet heat kernel for $M$; $u(x;t)=0$ for all $x\in \partial M$ and for all $t>0$.

Let $\rho :M \rightarrow \mathbb R$ be the specific heat of $M$; we suppose that $\rho$ is continuous. Following [1] we define the heat content of $\Omega$ in $M$ by setting

$$\begin{aligned} \beta _\Omega (\phi ,\rho ,\Delta _g)(t):=\int \limits _{\Omega }\int \limits _{\Omega }K(x,\tilde{x};t)\phi (x)\rho (\tilde{x})dxd\tilde{x}. \end{aligned}$$

(1.3)

Let $\phi$ and $\rho$ be smooth on $\Omega$ and let $(M,g)$ satisfy one of the conditions described in Sect. 1.1. Theorem 1.2 and Theorem 1.3 below show there exists a complete asymptotic series

$$\begin{aligned} \beta _\Omega (\phi ,\rho ,\Delta _g)(t)&\sim \sum _{n=0}^\infty t^{n}\beta _{n}^\Omega (\phi ,\rho ,\Delta _g) +\sum _{j=0}^\infty t^{(1+j)/2}\beta _{j}^{\partial \Omega }(\phi ,\rho ,\Delta _g)\text { as }t\rightarrow 0^+, \end{aligned}$$

where $\beta _{n}^\Omega$ and $\beta _{j}^{\partial \Omega }$ are the integrals of certain locally computable invariants over $\Omega$ and $\partial \Omega$, respectively. We extend $\phi$ by $0$ on $M\setminus \Omega$, and note that $\phi$ may be discontinuous on all or part of $\partial \Omega$. In that case the initial condition is satisfied for $x\in M\setminus \partial \Omega$.

The study of the heat content of $\Omega$ in $\mathbb R^m$ was initiated in [11]. It was shown [9–11] that if $\Omega$ is an open set in $\mathbb R^m$ with $C^{1,1}$ boundary $\partial \Omega$, with finite Lebesgue measure $|\Omega |$, and with finite perimeter $\mathcal P(\Omega )$ then

$$\begin{aligned} \mathcal P(\Omega )=\lim _{t\rightarrow 0}\left( \frac{\pi }{t}\right) ^{1/2}\int \int _{\Omega \times (\mathbb R^m\setminus \Omega )}K(x,\tilde{x};t)dxd\tilde{x}, \end{aligned}$$

where $K(x,\tilde{x};t)=(4\pi t)^{-m/2}e^{-|x-\tilde{x}|^2/(4t)}$ is the heat kernel for $\mathbb R^m$. It immediately follows that

$$\begin{aligned} \beta _\Omega (1,1,\Delta _{g_e})(t)=|\Omega |-\pi ^{-1/2}\mathcal P(\Omega )t^{1/2}+o(t^{1/2}), \ t\downarrow 0, \end{aligned}$$

(1.4)

where $|\Omega |=\int \limits _{\Omega }1dx$. We note that the main contribution beyond the constant term $|\Omega |$ in Eq. (1.4) comes from localization near $\partial \Omega$. We shall see presently in Theorem 5.1 that only the geometry near $\Omega$ plays a role in the heat content modulo an exponentially small error in $t^{-1}$ as $t\downarrow 0$ and which therefore plays no role in the asymptotic series. Even though the $t\downarrow 0$ behavior of the heat kernel is known for general $(M,g)$ [8], this explicit asymptotic behavior does not give much insight, and is not helpful in the determination of the locally computable invariants of $\Omega$ and of $\partial \Omega$, respectively.

We must employ a more general formalism. Even if we were only interested in the scalar Laplacian, it is a facet of the “method of universal examples” that one must examine this more general framework. Let $(M,g)$ be as described in Sect. 1.1. Let $D_M$ be an operator of Laplace type on a smooth vector bundle $V$ over a complete Riemannian manifold $(M,g)$. Let $\Omega$ be a compact $m$-dimensional submanifold of $M$ with smooth boundary. Let $\phi \in L^1(V|_{\Omega })$ represent the initial temperature, and let $\rho \in L^1(V^*|_{\Omega })$ represent the specific heat. As above, we extend $\phi$ and $\rho$ to $M$ to be zero on $\Omega ^c$, and we denote the resulting extensions by $\phi _\Omega$ and $\rho _\Omega$ to emphasize that they are supported on $\Omega$. Similarly to Eq. (1.3) we define the heat content of $\Omega$ in $M$ by

$$\begin{aligned} \beta _\Omega (\phi ,\rho ,{ D_M})(t)=\beta _M(\phi _{\Omega },\rho _{\Omega },D_M)(t)=\int \limits _\Omega \int \limits _\Omega \langle K(x,\tilde{x};t)\phi (x),\rho (\tilde{x})\rangle dxd\tilde{x}, \end{aligned}$$

where $\langle \cdot ,\cdot \rangle$ denotes the natural pairing between $V$ and $V^*$. As above, we shall suppose that both $\rho$ and $\phi$ are smooth on the interior of $\Omega$. However, we obtain additional information by permitting $\phi$ to have a controlled singularity near $\partial \Omega$. If $r$ is the geodesic distance to the boundary, we shall suppose that $r^{\alpha }\phi$ is smooth near the boundary for some fixed complex number $\alpha$. We shall always assume $\mathfrak {R}(\alpha )<1$ to ensure that $\phi \in L^1(\Omega )$. The parameter $\alpha$ controls the blow up (if $\mathfrak {R}(\alpha )>0$) or decay (if $\mathfrak {R}(\alpha )<0$) of $\phi$ near the boundary. We permit $\alpha$ to be complex. Although this has no physical significance, it is useful in analytic continuation arguments as we shall see in the proof of Lemma 3.1. Let $\mathcal {K}_\alpha =\mathcal {K}_\alpha (V)$ denote the resulting space of functions. We will establish the following result in Sect. 2:

Theorem 1.2

Adopt the notation established above. Let $M$ be a compact Riemannian manifold without boundary. There is a complete asymptotic series as $t\downarrow 0$ of the form:

$$\begin{aligned} \beta _\Omega (\phi ,\rho ,D_M)(t)&\sim \sum _{n=0}^\infty t^{n}\beta _{n}^\Omega (\phi ,\rho ,D_M) +\sum _{j=0}^\infty t^{(1+j-\alpha )/2}\beta _{j,\alpha }^{\partial \Omega }(\phi ,\rho ,D_M), \end{aligned}$$

where $\beta _{n}^\Omega$ and $\beta _{j,\alpha }^{\partial \Omega }$ are integrals of certain locally computable invariants over $\Omega$ and $\partial \Omega$, respectively.

In Sect. 5, we will use Theorem 1.2 to establish the following result:

Theorem 1.3

Let $(M,g)=(\mathbb {R}^m,g_e)$ or let $M$ be a compact submanifold of $\mathbb {R}^m$ with smooth boundary and $g=g_e|_M$. There is a complete asymptotic series as $t\downarrow 0$ of the form:

$$\begin{aligned} \beta _\Omega (\phi ,\rho ,D_M)(t)&\sim \sum _{n=0}^\infty t^{n}\beta _{n}^\Omega (\phi ,\rho ,D_M) +\sum _{j=0}^\infty t^{(1+j-\alpha )/2}\beta _{j,\alpha }^{\partial \Omega }(\phi ,\rho ,D_M), \end{aligned}$$

where $\beta _{n}^\Omega$ and $\beta _{j,\alpha }^{\partial \Omega }$ are integrals of certain locally computable invariants over $\Omega$ and $\partial \Omega$, respectively.

1.3 One-Dimensional Geometry

What happens on the line is in many ways crucial to our analysis, as we shall “bootstrap” our way from that setting to the higher-dimensional setting. Let $S^1=[0,2\pi ]$, where we identify $0\sim 2\pi$. Let $\Omega =[0,\pi ]\subset S^1$. Let $V=V^*$ be the trivial bundles, and let $D=-\partial _x^2$. Near the boundary point $x=0$, we may expand $\phi$ and $\rho$ in modified Taylor series for $x>0$:

$$\begin{aligned} \phi (x)\sim x^{-\alpha }\{\phi _0+\phi _1x+\phi _2x^2+\cdots \}\text { and }\rho (x)\sim \rho _0+\rho _1x+\rho _2x^2+\cdots . \end{aligned}$$

There is a similar expansion near $x=\pi$ as well. $\partial \Omega =\{0,\pi \}$ and $dy$ is simply counting measure in this instance. We will establish the following result in Sect. 3:

Lemma 1.4

With the notation established above, we may take

$$\begin{aligned} \beta _n^\Omega =\frac{(-1)^n}{n!}\int \limits _\Omega \phi \cdot D^n\rho dx,\text { and }\beta _{j,\alpha }^{\partial \Omega } =\sum _{k+\ell =j}\int \limits _{\partial \Omega }c_{k,\ell ,\alpha }\phi _k\rho _\ell dy, \end{aligned}$$

where $dy$ is the Riemannian volume element on the boundary of $\partial \Omega$, and where

$$\begin{aligned} c_{k,\ell ,\alpha }=(-1)^{\ell +1}\frac{1}{\sqrt{4\pi }}\int \limits _{0}^\infty \int \limits _{0}^\infty e^{-(u+\tilde{u})^2/4}u^{k-\alpha }\tilde{u}^\ell dud\tilde{u}\ . \end{aligned}$$

1.4 Local Invariants

We return to the general setting. By considering the case in which $\Omega =M$, we see that $\beta _n^\Omega$ can be taken to be

$$\begin{aligned} \beta _{n}^\Omega (\phi ,\rho ,{ D_M})=\frac{(-1)^n}{n!}\int \limits _\Omega \langle \phi , \tilde{D}_M^n\rho \rangle dx, \end{aligned}$$

where $\tilde{D}_M^n$ is the dual operator on $V^*$. We shall not differentiate $\phi$ on the interior owing to the lack of smoothness in $\phi$ as we approach the boundary. This is a crucial point: were we to examine the doubly singular setting, we would need to regularize the interior integrals. Thus attention is focused on the boundary invariants $\beta _{j,\alpha }^{\partial \Omega }$; these are only defined up to divergence terms.

1.5 A Bochner Formalism

Although in Sects. 2 and 3 we will use a coordinate formalism for describing the invariants $\beta _{j,\alpha }^{\partial \Omega }$, it is useful to introduce an invariant tensorial formalism at this point. There is a unique connection $\nabla$ on $V$ and a unique endomorphism $E$ of $V$ so that we may express $D_M$ using a Bochner formalism:

$$\begin{aligned} D_M\phi =-\{g^{ij}\phi _{;ij}+E\phi \}. \end{aligned}$$

Here $\phi _{;ij}$ denotes the components of the second covariant derivative $\nabla ^2\phi$ of $\phi$. We adopt the Einstein convention and sum over repeated indices (see Gilkey [6] for details). Let $\Gamma$ denote the Christoffel symbols of the metric $g$. If we express $D_M$ in a coordinate system in the form

$$\begin{aligned} \displaystyle D_M=-\{ g^{ij}\mathrm{Id }\partial _{x_i} \partial _{x_j}+A^i\partial _{x_i}+B\}, \end{aligned}$$

(1.5)

then the connection $1$-form $\omega$ of $\nabla$ and the endomorphism $E$ are given by

$$\begin{aligned} \omega _i=\textstyle {\textstyle \frac{1}{2}}(g_{ij}A^j +g^{kl}\Gamma _{kli}\mathrm{Id })\text { and } E=B-g^{ij}(\partial _{x_j}\omega _{i}+\omega _{i}\omega _{j} -\omega _{k}\Gamma _{ij}{}^k). \end{aligned}$$

The question always arises as to why it is necessary (or desirable) to consider these more general operators of Laplace type when in practice one is usually only interested in the scalar Laplacian. In Sect. 3.4, we will evaluate the coefficients of the terms involving the second fundamental form $L_{ab}$ and the Ricci curvature $\mathrm{Ric }_{mm}$ in $\beta _{1,\alpha }^{\partial \Omega }$ and $\beta _{2,\alpha }^{\partial \Omega }$. We will use warped products and operators which are not the scalar Laplacian. The fact that we are working with quite general operators will be crucial. It is typical in this subject that to obtain formulas for the scalar Laplacian, one must invoke the more general formalism and derive formulas in a very general context.

Let $\nabla ^*$ be the dual connection on $V^*$, and let the associated connection $1$-form be given by $-\omega _i^*$. We let indices $\{i,j,k,l\}$ range from $1$ to $m$. Let $R_{ijkl}$ be the components of the curvature tensor, let $\mathrm{Ric }_{ij}$ be the components of the Ricci tensor, and let $\tau$ be the scalar curvature. Let indices $\{a,b,c\}$ range from $1$ to $m-1$. Near the boundary, we choose a local orthonormal frame $\{e_i\}$ for the tangent bundle so that $e_m$ is the inward unit geodesic normal. Let $L_{ab}$ denote the components of the second fundamental form. Let

$$\begin{aligned} \phi _j:=\nabla _{e_m}^j\phi |_{\partial M}\text { and } \rho _j:=(\nabla ^*_{e_m})^j\rho |_{\partial M}. \end{aligned}$$

One may use dimensional analysis to see that $\beta _{j,\alpha }^{\partial \Omega }$ is homogeneous of order $j$ in the derivatives of the structures involved. This leads to the following observation.

Lemma 1.5

There exist universal constants $\varepsilon _{\nu ,\alpha }$ which depend holomorphically on $\alpha$ so that

$$\begin{aligned}&\beta _{0,\alpha }^{\partial \Omega } (\phi ,\rho , D_M)=\displaystyle \int \limits _{\partial \Omega } \varepsilon _{0,\alpha }\langle \phi _0,\rho _0\rangle dy. \\&\beta _{1,\alpha }^{\partial \Omega }(\phi ,\rho , D_M)=\displaystyle \int \limits _{\partial \Omega }\left\{ \varepsilon _{1,\alpha }\langle \phi _1,\rho _0\rangle +\varepsilon _{2,\alpha }\langle L_{aa}\phi _0,\rho _0\rangle +\varepsilon _{3,\alpha }\langle \phi _0,\rho _1\rangle \right\} dy. \\&\beta _{2,\alpha }^{\partial \Omega }(\phi ,\rho , D_M)=\displaystyle \int \limits _{\partial \Omega }\{\varepsilon _{4,\alpha }\langle \phi _2,\rho _0\rangle +\varepsilon _{5,\alpha }\langle L_{aa}\phi _1,\rho _0\rangle +\varepsilon _{6,\alpha }\langle E\phi _0,\rho _0\rangle \\&\quad +\,\varepsilon _{7,\alpha }\langle \phi _0,\rho _2\rangle +\varepsilon _{8,\alpha }\langle L_{aa}\phi _0,\rho _1\rangle +\varepsilon _{9,\alpha }\langle \mathrm{Ric }_{mm}\phi _0,\rho _0\rangle +\varepsilon _{10,\alpha }\langle L_{aa}L_{bb}\phi _0,\rho _0\rangle \\&\quad +\,\varepsilon _{11,\alpha }\langle L_{ab}L_{ab}\phi _0,\rho _0\rangle +\varepsilon _{12,\alpha }\langle \phi _{0;a},\rho _{0;a}\rangle +\varepsilon _{13,\alpha }\langle \tau \phi _0,\rho _0\rangle +\varepsilon _{14,\alpha }\langle \phi _1,\rho _1\rangle \}dy. \end{aligned}$$

We omit details in the interest of brevity and refer instead to the discussion in [4], where a similar result is established for the heat content asymptotics which arise from the Dirichlet realization.

Although we have in principle permitted vector-valued operators, this lack of commutativity plays no role at the $\beta _{2,\alpha }^{\partial \Omega }$ level and thus we restrict henceforth to scalar operators.

1.6 Normalizing Constants

Following the discussion in [4], we define

$$\begin{aligned} c_\alpha :=2^{1-\alpha }\Gamma \left( \frac{2-\alpha }{2}\right) \frac{1}{\sqrt{\pi }(\alpha -1)}. \end{aligned}$$

Because $s\Gamma (s)=\Gamma (s+1)$, we have the recursion relations

$$\begin{aligned} c_\alpha =-\frac{\alpha -3}{2(\alpha -1)(\alpha -2)}c_{\alpha -2}\quad \text {and}\quad c_{\alpha +1}=-\frac{\alpha -2}{2\alpha (\alpha -1)}c_{\alpha -1}. \end{aligned}$$

(1.6)

We set $\alpha =0$ to see

$$\begin{aligned} c_0=-\frac{2}{\sqrt{\pi }},\quad c_{-1}=-1,\quad c_{-2}=-\frac{8}{3\sqrt{\pi }}. \end{aligned}$$

1.7 Some Terms in the Asymptotic Series

In Sect. 3, we will establish the following result by determining the universal constants in Lemma 1.5. This together with Theorem 1.2 are the two main results of this paper.

Theorem 1.6

$$\begin{aligned}&\beta _{0,\alpha }^{\partial \Omega } (\phi ,\rho ,{D_M})=c_\alpha \textstyle \int \limits _{\partial \Omega } \textstyle \frac{1}{2}\langle \phi _0,\rho _0\rangle dy. \\&\beta _{1,\alpha }^{\partial \Omega } (\phi ,\rho ,D_M) =\textstyle c_{\alpha -1}\int \limits _{\partial \Omega }\left\{ \textstyle \frac{1}{2}\langle \phi _1,\rho _0\rangle +\textstyle \frac{\alpha }{4(1-\alpha )}\langle L_{aa}\phi _0,\rho _0\rangle +\textstyle \frac{1}{2(\alpha -1)}\langle \phi _0,\rho _1\rangle \right\} dy. \\&\beta _{2,\alpha }^{\partial \Omega } (\phi ,\rho ,{D_M}) =\textstyle c_{\alpha -2}\int \limits _{\partial \Omega }\{\textstyle \frac{1}{2}\langle \phi _2,\rho _0\rangle +\textstyle \frac{\alpha -1}{4(2-\alpha )}\langle L_{aa}\phi _1,\rho _0\rangle +\textstyle \frac{3-\alpha }{4(1-\alpha )(2-\alpha )}\langle E \phi _0,\rho _0\rangle \\&\quad +\textstyle \frac{1}{(1-\alpha )(2-\alpha )}\langle \phi _0,\rho _2\rangle -\textstyle \frac{\alpha +1}{4(2-\alpha )(1-\alpha )}\langle L_{aa}\phi _0,\rho _1\rangle -\textstyle \frac{1-\alpha }{8(2-\alpha )}\langle \mathrm{Ric }_{mm}\phi _0,\rho _0\rangle \\&\quad +\textstyle \frac{3\alpha -5}{16(2-\alpha )}\langle L_{aa}L_{bb}\phi _0,\rho _0\rangle -\textstyle \frac{1-\alpha }{8(2-\alpha )}\langle L_{ab}L_{ab}\phi _0,\rho _0\rangle \\&\quad -\textstyle \frac{3-\alpha }{4(1-\alpha )(2-\alpha )}\langle \phi _{0;a},\rho _{0;a}\rangle +0\langle \tau \phi _0,\rho _0\rangle + \textstyle \frac{1}{2(\alpha -2)}\langle \phi _1,\rho _1\rangle \}dy. \end{aligned}$$

We specialize Theorem 1.6 to the smooth setting by setting $\alpha =0$ to obtain the following.

Corollary 1.7

$$\begin{aligned}&\beta _{0}^{\partial \Omega } (\phi ,\rho , D_M)=-\frac{2}{\sqrt{\pi }}\textstyle \int \limits _{\partial \Omega } \textstyle \frac{1}{2}\langle \phi _0,\rho _0\rangle dy. \\&\beta _{1}^{\partial \Omega } (\phi ,\rho , D_M) =\textstyle -\int \limits _{\partial \Omega }\left\{ \textstyle \frac{1}{2}\langle \phi _1,\rho _0\rangle +0\langle L_{aa}\phi _0,\rho _0\rangle -\textstyle \frac{1}{2}\langle \phi _0,\rho _1\rangle \right\} dy. \\&\beta _{2}^{\partial \Omega } (\phi ,\rho , D_M) =\textstyle -\frac{8}{3\sqrt{\pi }}\int \limits _{\partial \Omega }\{\textstyle \frac{1}{2}\langle \phi _2,\rho _0\rangle -\textstyle \frac{1}{8}\langle L_{aa}\phi _1,\rho _0\rangle +\textstyle \frac{3}{8}\langle E \phi _0,\rho _0\rangle \\&\quad +\,\textstyle \frac{1}{2}\langle \phi _0,\rho _2\rangle -\textstyle \frac{1}{8}\langle L_{aa}\phi _0,\rho _1\rangle -\textstyle \frac{1}{16}\langle \mathrm{Ric }_{mm}\phi _0,\rho _0\rangle -\textstyle \frac{5}{32}\langle L_{aa}L_{bb}\phi _0,\rho _0\rangle \\&\quad \,-\textstyle \frac{1}{16}\langle L_{ab}L_{ab}\phi _0,\rho _0\rangle -\textstyle \frac{3}{8}\langle \phi _{0;a},\rho _{0;a}\rangle +0\langle \tau \phi _0,\rho _0\rangle -\textstyle \frac{1}{4}\langle \phi _1,\rho _1\rangle \}dy. \end{aligned}$$

1.8 Dirichlet and Robin Boundary Conditions

Let $\mathcal {B}_Df:=f|_{\partial \Omega }$ be the Dirichlet boundary operator and let $\mathcal {B}_Sf:=(\nabla _mf+Sf)|_{\partial \Omega }$ be the Robin boundary operator; here $S$ is an auxiliary endomorphism of $V|_{\partial \Omega }$. We use $S^*$ and the dual connection to define the dual boundary operator $\tilde{\mathcal {B}}$ on $C^\infty (V^*)$. Let $D_{\mathcal {B}}$ denote the Dirichlet or Robin realization of $D_M$. In this instance, the ambient manifold $M$ plays no role and defines

$$\begin{aligned} \beta (\phi ,\rho , D_M,\mathcal {B})=\int \limits _\Omega \langle e^{-tD_{\mathcal {B}}}\phi ,\rho \rangle dx. \end{aligned}$$

The existence of an analogous asymptotic series in this setting has been established in [4]. After adjusting the notation suitably using Eq. (1.6) from that in [4], the results of [4] yield the following.

Theorem 1.8

• (1) $\beta _{0,\alpha }^{\partial M}(\phi ,\rho ,D_M,{B_{\mathcal {D}}})=c_\alpha \int \limits _{\partial \Omega }\langle \phi _0,\rho _0\rangle dy.$

• (2) $\beta _{1,\alpha }^{\partial M}(\phi ,\rho ,D_M,{B_{\mathcal {D}}})=c_{\alpha -1}\int \limits _{\partial \Omega } \langle \phi _1-{\textstyle {\frac{1}{2}}}L_{aa}\phi _0,\rho _0\rangle dy.$

• (3) $\beta _{2,\alpha }^{\partial M}(\phi ,\rho ,D_M,{B_{\mathcal {D}}}) =c_{\alpha -2}\int \limits _{\partial \Omega }\{\langle \phi _2,\rho _0\rangle -\frac{1}{2}\langle L_{aa}\phi _1,\rho _0\rangle$

  $\quad -\frac{\alpha -3}{2(\alpha -1)(\alpha -2)}\langle E\phi _0,\rho _0\rangle +\frac{2}{(\alpha -1)(\alpha -2)}\langle \phi _0,\rho _2\rangle -\frac{1}{(\alpha -1)(\alpha -2)}\langle L_{aa}\phi _0,\rho _1\rangle$

  $\quad +\frac{\alpha -3}{2(\alpha -1)(\alpha -2)}\langle \phi _{0:a},\rho _{0:a}\rangle -\frac{\alpha -1}{4(\alpha -2)}\langle \mathrm{Ric }_{mm}\phi _0,\rho _0\rangle$

  $\quad +\frac{\alpha -1}{8(\alpha -2)}\langle L_{aa}L_{bb}\phi _0,\rho _0\rangle -\frac{\alpha -1}{4(\alpha -2)} \langle L_{ab}L_{ab}\phi _0,\rho _0\rangle \}dy.$

• (4) $\beta _{0,\alpha }^{\partial M}(\phi ,\rho ,D_M,{B_{\mathcal {R}}})=0.$

• (5) $\beta _{1,\alpha }^{\partial M}(\phi ,\rho , D_M,{B_{\mathcal {R}}})=\frac{2\alpha }{2-\alpha }c_{\alpha +1}\int \limits _{\partial M}\langle \phi _0,\tilde{B}_{\mathcal {R}}\rho \rangle dy$

  $\quad =-\frac{2\alpha }{2-\alpha }\cdot \frac{\alpha -2}{2\alpha (\alpha -1)}c_{\alpha -1}\int \limits _{\partial \Omega } \langle \phi _0,\tilde{B}_{\mathcal {R}}\rho \rangle dy$

  $\quad =-\frac{1}{1-\alpha }c_{\alpha -1}\int \limits _{\partial \Omega } \langle \phi _0,\tilde{B}_{\mathcal {R}}\rho \rangle dy.$

• (6) $\beta _{2,\alpha }^{\partial M}(\phi ,\rho , D_M,{B_{\mathcal {R}}}) =\frac{-2}{3-\alpha }c_\alpha \int \limits _{\partial M}\langle (1-\alpha ) \phi _1+S\phi _0-\frac{\alpha }{2}L_{aa}\phi _0,\tilde{B}_{\mathcal {R}}\rho \rangle dy$

  $\quad =-\frac{1}{(\alpha -1)(\alpha -2)}c_{\alpha -2}\int \limits _{\partial M}\langle (1-\alpha ) \phi _1+S\phi _0-\frac{\alpha }{2}L_{aa}\phi _0,\tilde{B}_{\mathcal {R}}\rho \rangle dy.$

2 Proof of Theorem 1.2

We will follow the discussion of the pseudo-differential calculus based on the work by Seeley [12, 13], and refer to Gilkey [6]. Throughout this section, we assume the ambient Riemannian manifold $(M,g)$ is compact and without boundary.

By using a partition of unity, we may assume that $\rho$ and $\phi$ are supported within coordinate systems. Since the kernel of the heat equation decays exponentially in $t^{-1}$ for $\mathrm{dist }_g(x,\tilde{x})\ge \epsilon >0$, we may assume that $\rho$ and $\phi$ have support within the same coordinate system. There will, of course, be three different types of coordinate systems to be considered—those which touch the boundary of $\Omega$, those which are contained entirely within the interior of $\Omega$, and those which are contained in the exterior of $\Omega$; those contained in the exterior of $\Omega$ play no role as they contribute an exponentially small error in $t^{-1}$. In Sect. 2.1 we establish notational conventions and prove a technical result; in Sect. 2.2 we review the notion of a pseudo-differential operator; in Sect. 2.3, we construct the resolvent; and in Sect. 2.4 we construct an approximation to the kernel of the heat equation. The new material begins in Sect. 2.5, where we begin the examination of the heat content. In Sect. 2.6, we establish the existence of two different kinds of asymptotic series. In Sect. 2.7, we use the results of Sect. 2.6 to discuss coordinate systems contained in the interior of $\Omega$, and in Sect. 2.8 we use the results of Sect. 2.6 to study coordinate systems near the boundary. The fact that $D_M$ is of Laplace type plays a central role in the discussion.

2.1 Notational Conventions

Let $x=(x^1,\ldots ,x^m)\in \mathbb {R}^m$ be coordinates on an open set $\mathcal {O}\subset M$. Let $(x,\xi )$ be the induced coordinate system on the cotangent space $T^*(\mathcal {O})$, where we expand a $1$-form $\omega \in T^*(\mathcal {O})$ in the form

$$\begin{aligned} \omega =\xi _idx^i\text { to define }\xi =(\xi _1,\ldots ,\xi _m). \end{aligned}$$

We let $x\cdot \xi$ be the natural pairing

$$\begin{aligned} x\cdot \xi :=x^i\xi _i. \end{aligned}$$

If $\alpha =(a_1,\ldots ,a_m)$ is a multi-index, set

$$\begin{aligned} \begin{array}{llll} |\alpha |=a_1+\cdots +a_m,&{}\alpha !=a_1!\cdots a_m!,&{}\partial _x^\alpha :=\partial _{x_1}^{a_1}\cdots \partial _{x_m}^{a_m},\\ \phi ^{(\alpha )}:=\partial _x^\alpha \phi ,&{} \rho ^{(\alpha )}:=\partial _x^\alpha \rho ,&{}D_x^\alpha :=\sqrt{-1}^{|\alpha |}d_x^\alpha ,\\ d_\xi ^\alpha =\partial _{\xi _1}^{a_1}\cdots \partial _{\xi _m}^{a_m},&{}x^\alpha =x_1^{a_1}\cdots x_m^{a_m},&{} \xi ^\alpha :=\xi _1^{a_1}\cdots \xi _m^{a_m}. \end{array} \end{aligned}$$

For example, with these notational conventions, Taylor’s theorem becomes

$$\begin{aligned} f(x)=\sum _{|\alpha |\le n}\textstyle \frac{1}{\alpha !}f^{(\alpha )}(x_0)(x-x_0)^\alpha +O(|x-x_0|^{n+1}). \end{aligned}$$

(2.1)

Let $d\nu _x$, $d\nu _{\tilde{x}}$, and $d\nu _\xi$ denote Lebesgue measure on $\mathbb {R}^m$. Let $L^2_e$ denote $L^2(\mathbb {R}^m)$ with respect to Lebesgue measure. Let $g(x)=g_{ij}(x)dx^i\circ dx^j$ be a Riemannian metric on $\mathcal {O}$ and define

$$\begin{aligned} ||\xi ||_{g^*(x)}^2:=g^{ij}(x)\xi _i\xi _j\text { and }||x-\tilde{x}||_{g(x)}^2:=g_{ij}(x)(x^i-\tilde{x}^i)(x^j-\tilde{x}^j). \end{aligned}$$

We shall always be restricting to compact $x$ and $\tilde{x}$ subsets. Let $(\cdot ,\cdot )_e$ and $||_e$ denote the usual Euclidean inner product and norm, respectively:

$$\begin{aligned} (x,y)_e:=x_1y_1+\cdots +x_my_m\text { and } ||(x_1,\ldots ,x_m)||_e^2:=(x,x)_e=x_1^2+\cdots +x_m^2. \end{aligned}$$

We have estimates

$$\begin{aligned} C_1||\xi ||_e^2\le ||\xi ||_{g^*(x)}^2\le C_2||\xi ||_e^2 \text { and }C_1||x-\tilde{x}||_e^2\le ||x-\tilde{x}||_{g(x)}^2\le C_2||x-\tilde{x}||_e^2 \end{aligned}$$

for some constants $C_1>0$ and $C_2>0$. Let $\theta =\sqrt{g}$; $\theta$ is a symmetric metric so that $\theta _{ij}\theta _{jk}=g_{ik}$. We then have

$$\begin{aligned} ||\xi ||_{g^*(x)}^2=||\theta ^{-1}(x)\xi ||_e^2\text { and }||(x-\tilde{x})||_{g(x)}^2=||\theta (x)(x-\tilde{x})||_e^2. \end{aligned}$$

Lemma 2.1

We use $\theta ^2=g$ to lower indices and regard $\theta ^2(x)(x-\tilde{x})$ as a vector $(X-\tilde{X})_i:=g_{ij}(x-\tilde{x})^j$. With this identification, we have

$$\begin{aligned}&||\xi ||_{g^*(x)}^2+\sqrt{-1}(x-\tilde{x})\cdot \xi /\sqrt{t}\\&\quad =||\xi +\textstyle \frac{1}{2}\sqrt{-1}(X-\tilde{X})/\sqrt{t}||_{g^*(x)}^2+||(x-\tilde{x})||_{g(x)}^2/(4t). \end{aligned}$$

Proof

We expand:

$$\begin{aligned}&||\xi ||_{g^*(x)}^2+\sqrt{-1}(x-\tilde{x})\cdot \xi /\sqrt{t} \\&\quad =(\theta ^{-1}(x)\xi ,\theta ^{-1}(x)\xi )_e+\sqrt{-1}(\theta ^{-1}(x)\theta ^2(x)(\tilde{x}-x)/\sqrt{t},\theta ^{-1}(x)\xi )_e \\&\quad =(\theta ^{-1}(x)\{\xi +\textstyle \sqrt{-1}\theta ^{2}(x)(\tilde{x}-x)/\sqrt{t}\},\theta ^{-1}(x)\xi )_e \\&\quad =||\theta ^{-1}(x)\{\xi +\frac{1}{2}\sqrt{-1}\theta ^{2}(x)(\tilde{x}-x)/\sqrt{t}\}||_e^2 +||\theta (x)(\tilde{x}-x)||_e^2/(4t) \\&\quad =||\xi +\frac{1}{2}\sqrt{-1}(X-\tilde{X})/\sqrt{t}||_{g^*(x)}^2+||(x-\tilde{x})||_{g(x)}^2/(4t). \end{aligned}$$

$\square$

2.2 Pseudo-Differential Operators

If $P$ is a pseudo-differential operator with symbol $p(x,\xi )$, then $P$ is characterized by the following identity for all $\phi \in C_0^\infty (V)$ and $\rho \in C_0^\infty (V^*)$:

$$\begin{aligned} \langle P\phi ,\rho \rangle _{L_e^2}=(2\pi )^{-m} \int \!\!\int \!\!\int e^{-\sqrt{-1}(x-\tilde{x})\cdot \xi }\langle p(x,\xi )\phi (x),\rho (\tilde{x})\rangle d\nu _xd\nu _\xi d\nu _{\tilde{x}}. \end{aligned}$$

(2.2)

The integrals in question here are iterated integrals—the convergence is not absolute and the $d\nu _x$ integral has to be performed before the $d\nu _\xi$ integral. However, if $p(x,\xi )$ decays rapidly enough in $\xi$, then the integrals are in fact absolutely convergent and we can interchange the order of integration to see following [6, Lemma 1.2.5] that $P$ is given by a kernel:

$$\begin{aligned} \begin{array}{l} \displaystyle \langle P\phi ,\rho \rangle _{L^2} =\int \langle K_P(x,\tilde{x})\phi (x),\rho (\tilde{x})\rangle d\nu _xd\nu _{\tilde{x}} \text { where}\\ \displaystyle K_P(x,\tilde{x}):=(2\pi )^{-m}\int e^{-\sqrt{-1}(x-\tilde{x})\cdot \xi }p(x,\xi )d\nu _\xi . \end{array} \end{aligned}$$

(2.3)

2.3 The Resolvent

Let $D_M$ be an operator of Laplace type on $C^\infty (V)$ over $M$. In a system of local coordinates $(x^1,\ldots ,x^m)$ on an open subset $\mathcal {O}$ of $M$, we may change notation slightly from that employed in Eq. (1.5) and expand:

$$\begin{aligned} D_M=a_2^{ij}(x)D_{x_i}D_{x_j}+a_1^i(x)D_{x_i}+a_0(x). \end{aligned}$$

We ensure that Eq. (2.2) defines the operator $D_M$ by defining:

$$\begin{aligned}&p(x,\xi )=p_2(x,\xi )+p_1(x,\xi )+p_0(x,\xi )\text { where }\\&p_2(x,\xi ):=|\xi |_{g^*(x)}^2,\quad p_1(x,\xi ):=a_1^i(x)\xi _i,\quad p_0(x,\xi ):=a_0(x). \end{aligned}$$

Let $\mathcal {R}\subset \mathbb {C}$ be the complement of a cone of angle $\varepsilon _{1,\alpha }$ about the positive real axis and a ball of radius $\varepsilon _{2,\alpha }^{-1}$ about the origin, where $\varepsilon _{2,\alpha }=\varepsilon _{2,\alpha }(\varepsilon _{1,\alpha })$ is chosen so that $D_M$ has no eigenvalues in $\mathcal {R}$. We let $\lambda \in \mathcal {R}$ henceforth. Following the discussion of [6, Lemma 1.7.2], we define $r_n(x,\xi ;\lambda )$ for $(x,\xi )\in T^*(\mathcal {O})$ and $\lambda \in \mathcal {R}$ inductively by setting

$$\begin{aligned} \begin{array}{l} r_0(x,\xi ;\lambda ):=(|\xi |^2_{g^*(x)}-\lambda )^{-1},\\ r_n:=-r_0\displaystyle \sum _{|\alpha |+2+j-k=n,j<n} {\textstyle \frac{1}{\alpha !}}\partial _\xi ^\alpha p_k\cdot D_x^\alpha r_j\text { for }n>0. \end{array} \end{aligned}$$

(2.4)

Define

$$\begin{aligned} \begin{array}{lll} \mathrm{ord }(\partial _x^\alpha p_2)=|\alpha |,&{}\mathrm{ord }(\partial _x^\alpha p_1)=|\alpha |+1,&{} \mathrm{ord }(\partial _x^\alpha p_0|)=|\alpha |+2,\\ \mathrm{weight }(\lambda )=2,&{}\mathrm{weight }(\xi )=1. \end{array} \end{aligned}$$

The following lemma follows immediately by induction from the recursive definition in Eq. (2.4).

Lemma 2.2

• (1) $r_n$ is homogeneous of order $n$ in the derivatives of the symbol of $D_M$.

• (2) $r_n$ has weight $-n-2$ in $(\xi ,\lambda )$.

• (3) There exist polynomials $r_{n,j,\alpha }(x,D_M)$ for $n\le j\le 3n$ which are homogeneous of order $n$ in the derivatives of the symbol of $D_M$, so

  $$\begin{aligned} r_n(x,\xi ;\lambda )=\sum _{2j-|\alpha |=n}r_{n,j,\alpha }(x,D_M)(|\xi |_{g^*(x)}^2-\lambda )^{-j-1}\xi ^\alpha . \end{aligned}$$

We use Eq. (2.2) to define the pseudo-differential operator $R_n(\lambda )$ with symbol $r_n$ so that

$$\begin{aligned} \langle R_n(\lambda )\phi ,\rho \rangle _{L^2_e}=(2\pi )^{-m}\int \!\!\int \!\!\int e^{-\sqrt{-1}(x-\tilde{x})\cdot \xi }\langle r_n(x,\xi ;\lambda )\phi (x), \rho (\tilde{x})\rangle d\nu _xd\nu _\xi d\nu _{\tilde{x}}. \end{aligned}$$

Let $||\cdot ||_{-k,k}$ be the norm of a map from the Sobolev space $H_{-k}$ to the Sobolev space $H_k$. By [6, Lemma 1.7.3] we have that if $\lambda \ge \lambda (k)$ and if $n\ge n(k)$, then

$$\begin{aligned} ||(D_M-\lambda )^{-1}-R_0(\lambda )-\cdots -R_n(\lambda )||_{-k,k}\le C_k(1+|\lambda |)^{-k}. \end{aligned}$$

2.4 An Approximation to the Kernel of the Heat Equation

Let $\gamma$ be the boundary of $\mathcal {R}$ oriented suitably. We use the operator-valued Riemann integral to define

$$\begin{aligned} e^{-tD_M}:=\frac{1}{2\pi \sqrt{-1}}\int \limits _\gamma e^{-t\lambda }(D_M-\lambda )^{-1}d\lambda . \end{aligned}$$

We use [6, Lemma 1.7.5] to see that this is the fundamental solution of the heat equation and belongs to $\mathrm{Hom }(H_{-k},H_k)$ for any $k$. We now let

$$\begin{aligned} e_n(x,\xi ;t)=\frac{1}{2\pi \sqrt{-1}}\int \limits _\gamma e^{-t\lambda }r_n(x,\xi ;\lambda )d\lambda \end{aligned}$$

(2.5)

define the pseudo-differential operator

$$\begin{aligned} E_n(t,D_M):=\frac{1}{2\pi \sqrt{-1}}\int \limits _\gamma e^{-t\lambda }R_n(\lambda )d\lambda . \end{aligned}$$

(2.6)

We use Lemma 2.2 (3) and Cauchy’s integral formula to rewrite Eq. (2.5) as

$$\begin{aligned} e_n(x,\xi ;t)=\sum _{2j-|\alpha |=n}\frac{t^j}{j!}\xi ^\alpha e^{-t|\xi |_{g^*(x)}^2}r_{n,j,\alpha }(x,D_M). \end{aligned}$$

(2.7)

We now use Eqs. (2.3) and (2.7) to see that the operator $E_n$ of Eq. (2.6) is given by the smooth kernel

$$\begin{aligned}&K_n(x,\tilde{x};t)\\&\quad :=\sum _{2j-|\alpha |=n}(2\pi )^{-m}\frac{t^j}{j!}\int \limits _{\mathbb {R}^m} e^{-t|\xi |_{g^*(x)}^2-\sqrt{-1}(x-\tilde{x})\cdot \xi }\xi ^\alpha r_{n,j,\alpha }(x,D_M)d\nu _\xi .\nonumber \end{aligned}$$

(2.8)

Let $||\cdot ||_{C^k}$ denote the $C^k$ norm. Given any $k\in \mathbb {N}$, there exists $n(k)$ so that if $n\ge n(k)$ and if $0<t<1$, then [6, Lemma 1.8.1] implies

$$\begin{aligned} ||e^{-tD_M}-\sum _{n=0}^{n(k)}E_n(t,D_M)||_{-k,k}\le C_kt^k. \end{aligned}$$

This gives rise to a corresponding estimate (after increasing $n(k)$ appropriately):

$$\begin{aligned} ||K(t,x,\tilde{x},D_M)-\sum _{n=0}^{n(k)}K_n(t,x,\tilde{x}, D_M)||_{C^k}\le C_kt^k. \end{aligned}$$

(2.9)

2.5 Examining the Heat Content

We use Eqs. (2.7), (2.8), and (2.9) to expand

$$\begin{aligned}&\beta (\phi ,\rho ,D_M)(t)=\sum _{2j-|\alpha |=0}^{n(k)}(2\pi )^{-m}\frac{t^j}{j!}\int \!\!\int \!\!\int e^{-t|\xi |_{g^*(x)}^2-\sqrt{-1}(x-\tilde{x})\cdot \xi }\xi ^\alpha \\&\qquad \qquad \qquad \qquad \times \langle r_{n,j,\alpha }(x,D_M)\phi (x),\rho (\tilde{x})\rangle d\nu _xd\nu _\xi d\nu _{\tilde{x}}+O(t^k). \end{aligned}$$

We examine a typical term in the sum, setting

$$\begin{aligned} \beta _{n,j,\alpha }(\phi ,\rho )(t)&:= (2\pi )^{-m}\frac{t^j}{j!}\int \!\!\int \!\!\int e^{-t|\xi |_{g^*(x)}^2-\sqrt{-1}(x-\tilde{x})\cdot \xi }\xi ^\alpha \\&\quad \times \langle r_{n,j,\alpha }(x)\phi (x),\rho (\tilde{x})\rangle d\nu _xd\nu _\xi d\nu _{\tilde{x}}. \end{aligned}$$

Here all integrals are over $\mathbb {R}^m$ and converge absolutely for $t>0$; $\phi$ and $\rho$ have compact support. We change variables, setting $\tilde{\xi }:=t^{1/2}\xi$ to express

$$\begin{aligned}&\beta _{n,j,\alpha }(\phi ,\rho )(t)=\frac{t^{j-\frac{1}{2}m-\frac{1}{2}|\alpha |}}{j!}(2\pi )^{-m} \int \!\!\int \!\!\int e^{-|\tilde{\xi }|_{g^*(x)}^2-\sqrt{-1}(x-\tilde{x})\cdot \tilde{\xi }/\sqrt{t}}\tilde{\xi }^\alpha \\&\qquad \qquad \qquad \qquad \times \displaystyle \langle r_{n,j,\alpha }(x,D_M)\phi (x),\rho (\tilde{x})\rangle d\nu _xd\nu _{\tilde{\xi }}d\nu _{\tilde{x}}. \end{aligned}$$

Note that $\frac{1}{2}n=j-\frac{1}{2}|\alpha |$. We adopt the notation of Lemma 2.1 and make a complex change of coordinates setting:

$$\begin{aligned} \textstyle \eta =\tilde{\xi }+\frac{1}{2}\sqrt{-1}(X-\tilde{X})/\sqrt{t}. \end{aligned}$$

We then apply Lemma 2.1 and the binomial theorem to express

$$\begin{aligned}&{\beta _{n,j,\alpha }(\phi ,\rho )(t)}\\&= (2\pi )^{-m}\sum _{\alpha _1+\alpha _2=\alpha } {\textstyle \frac{\alpha !}{j!\alpha _1!\alpha _2!}}(-\sqrt{-1})^{|\alpha _2|} t^{(n-m)/2}\int \!\!\int \!\!\int e^{-||\eta ||_{g^*(x)}^2}\eta ^{\alpha _1}\\&\qquad \times e^{-||x-\tilde{x}||_{g(x)}^2/(4t)}\left( \textstyle \frac{X-\tilde{X}}{2\sqrt{t}}\right) ^{\alpha _2} \langle r_{n,j,\alpha }(x,D_M)\phi (x),\rho (\tilde{x})\rangle d\nu _\eta d\nu _xd\nu _{\tilde{x}}. \end{aligned}$$

The $d\nu _\eta$ integral is over the complex domain $\eta \in \mathbb {R}+\frac{1}{2}\sqrt{-1}\textstyle \frac{X-\tilde{X}}{\sqrt{t}}$. But we can deform that domain back to the real domain $\eta \in \mathbb {R}$. Set

$$\begin{aligned}&c_{\alpha _1,\alpha _2,j}:=(2\pi )^{-m}\frac{1}{j!}\frac{(\alpha _1+\alpha _2)!}{\alpha _1!\alpha _2!} (-\sqrt{-1})^{|\alpha _2|} \int \eta ^{\alpha _1}e^{-||\eta ||_{g^*(x)}^2}d\nu _\eta \text { to express}\\&\beta _{n,j,\alpha }(t)=\sum _{\alpha _1+\alpha _2=\alpha }c_{\alpha _1,\alpha _2,j}t^{(n-m)/2}\\&\qquad \times \int \int e^{-||x-\tilde{x}||_{g(x)}^2/(4t)}\left( \textstyle \frac{X-\tilde{X}}{2\sqrt{t}}\right) ^{\alpha _2}\langle r_{n,j,\alpha }(x,D_M)\phi (x),\rho (\tilde{x})\rangle d\nu _xd\nu _{\tilde{x}}. \end{aligned}$$

This sum ranges over $|\alpha _1|$ even as otherwise $c_{\alpha _1,\alpha _2,j}$ vanishes. Thus $|\alpha _2|\equiv |\alpha |\equiv n\mod 2$. This reduces the proof to considering expressions of the form

$$\begin{aligned} f_{n,j,\alpha ,\alpha _2}(t)&:= t^{(n-m)/2}\int \int e^{-||x-\tilde{x}||_{g(x)}^2/(4t)}(\textstyle \frac{X-\tilde{X}}{\sqrt{t}})^{\alpha _2}\nonumber \\&\qquad \times \langle r_{n,j,\alpha }(x,D_M)\phi (x),\rho (\tilde{x})\rangle d\nu _xd\nu _{\tilde{x}}, \\&\text { where }|\alpha _2|\equiv n\text { mod }2\text { and }\mathrm{ord }(r_{n,j,\alpha }(x,D_M))=n.\nonumber \end{aligned}$$

(2.10)

2.6 Asymptotic Series

Before proceeding further with our analysis of Eq. (2.10), we must establish the existence of asymptotic series in certain quite general contexts:

Lemma 2.3

Let $\Phi \in L^1(\mathbb {R}^m)$, let $\rho \in C^\infty (\mathbb {R}^m)$ have compact support in an open subset $\mathcal {O}\subset \mathbb {R}^m$, and let $(X-\tilde{X})_i:=g_{ij}(x-\tilde{x})^j$. Let

$\displaystyle F(t):=t^{(n-m)/2}\int \limits _{\mathcal {O}}\int \limits _{\mathcal {O}} e^{-||x-\tilde{x}||^2_{g(x)}/(4t)}\left( {\textstyle \frac{X-\tilde{X}}{\sqrt{t}}}\right) ^{\alpha _2} \langle \Phi (x),\rho (\tilde{x})\rangle d\nu _xd\nu _{\tilde{x}},\text { if }m\ge 1$,

$\displaystyle G(t):=t^{(n-1)/2}\int \limits _0^{\infty }\int \limits _0^{\infty } e^{-||x+\tilde{x}||^2_e/(4t)}\left( {\textstyle \frac{X+\tilde{X}}{\sqrt{t}}}\right) ^{\alpha _2}\langle \Phi (x),\rho (\tilde{x})\rangle d\nu _xd\nu _{\tilde{x}},\text { if }m=1$.

• (1) There exist smooth coefficients $c_{\sigma ,\alpha _2}=c_{\sigma ,\alpha _2}(g(x))$ so that there is a complete asymptotic series as $t\downarrow 0^+$ of the form

  $$\begin{aligned} F(t)\sim \sum _{|\sigma |=0}^\infty t^{(n+|\sigma |)/2}\int \limits _{\mathcal {O}} c_{\sigma ,\alpha _2} \langle \Phi (x),\rho ^{(\sigma )}(x)\rangle d\nu _x. \end{aligned}$$

• (2) Let $m=1$. Near $0$, we suppose $\Phi \sim x^{-\alpha }\sum _{i\ge 0}C_ix^i$ for $\mathfrak {R}(\alpha )<1$. There exist universal constants $c_{i,j,\alpha _2}$ so that there is a complete asymptotic series as $t\downarrow 0^+$ of the form

  $$\begin{aligned} G(t)\sim \sum _{i,j=0}^\infty t^{(n+1+i+j-\alpha )/2}c_{i,j,\alpha _2} \langle C_i,\rho ^{(j)}(0)\rangle . \end{aligned}$$

Proof

We make the change of variables $\tilde{x}=x+u$ and dually $\tilde{X}=X+U$ where $U_i=g_{ij}u^j$ to express

$$\begin{aligned} F(t)=t^{(n-m)/2}\int \int e^{-||u||_{g(x)}^2/(4t)}\textstyle \left( \frac{U}{\sqrt{t}}\right) ^{\alpha _2}\langle \Phi (x),\rho (x+u)\rangle d\nu _ud\nu _x. \end{aligned}$$

The $d\nu _u$ integral decays exponentially for $|u|>t^{1/4}$ so we may assume the $d\nu _u$ integral is localized to $|u|<t^{1/4}$. For $u$ small, we use Eq. (2.1) to express

$$\begin{aligned}&\rho (x+u)\sim \sum _{|\sigma |\le N}\textstyle \frac{1}{\sigma !}u^\sigma d_x^\sigma \rho (x)+O(|u|^N),\\&F(t)=t^{(n-m)/2}\sum _{|\sigma |\le N}{\textstyle \frac{1}{\sigma !}}\int \int e^{-||u||_{g(x)}^2/(4t)}\\&\qquad \times \left\{ ({\textstyle \frac{U}{\sqrt{t}}})^{\alpha _2} u^\sigma \langle \Phi (x),\rho ^{(\sigma )}(x)\rangle +O(|u|^N)\right\} d\nu _ud\nu _x. \end{aligned}$$

We set $\tilde{u}=u/\sqrt{t}$ and $\tilde{U}=U/\sqrt{t}$ to express

$$\begin{aligned} F(t)\sim \sum _{|\sigma |\le N}{\textstyle \frac{1}{\sigma !}}t^{(n+|\sigma |)/2} \int \int e^{-||\tilde{u}||_{g(x)}^2/4}\tilde{U}^{\alpha _2}\tilde{u}^\sigma \langle \Phi (x),\rho ^{(\sigma )}(x)\rangle d\nu _{\tilde{u}}d\nu _x. \end{aligned}$$

The $d\nu _x$ integral remains an integral over $\mathcal {O}$. But as $t\downarrow 0$, the $d\nu _{\tilde{u}}$ integral expands to $\mathbb {R}^m$ and defines the coefficients $c_{\sigma ,\alpha _2}=c_{\sigma ,\alpha _2}(g)$. This establishes assertion (1).

Let $m=1$. We note that $G$ decays exponentially for $x\ge \varepsilon >0$ or $\tilde{x}\ge \varepsilon >0$. On the small square, we expand

$$\begin{aligned} \Phi (x)\sim \sum _{i\ge 0}C_ix^i\text { and }\rho (\tilde{x})\sim \tilde{x}^j\rho ^{(j)}(0). \end{aligned}$$

We then make the change of variables with $u=x/\sqrt{t}$ and $\tilde{u}=\tilde{x}/\sqrt{t}$ to express

$$\begin{aligned} G(t)\sim t^{(n+1)/2}\sum _{i+j\le N}t^{(i+j-\alpha )/2}c_{i,j,\alpha }\langle C_i,{\rho ^{(j)}}(0)\rangle \end{aligned}$$

where

$$\begin{aligned} \displaystyle c_{i,j,\alpha }:=\int \limits _0^\infty \int \limits _0^\infty e^{-\frac{1}{4}||u+\tilde{u}||^2_e}u^{i-\alpha }\tilde{u}^{j} d\nu _ud\nu _{\tilde{u}}. \end{aligned}$$

$\square$

2.7 The Interior Terms in Theorem 1.2

We apply Lemma 2.3 (1) to the case $\Phi =r_{n,j,\alpha }\phi$ in Eq. (2.10). By assumption, $r_{n,j,\alpha }$ is of order $n$ in the derivatives of the total symbol of $D_M$. We have that $\rho ^{(\sigma )}$ is of order $|\sigma |$ in the derivatives of $\rho$. Thus we have expressions which are of order $n+|\sigma |$ in the derivatives of the symbol of $D_M$ and in the derivatives of $\rho$. Furthermore, the $d\nu _{\tilde{u}}$ integral vanishes unless $|\sigma |+|\alpha _2|$ is even. Since $|\alpha _2|\equiv n$ mod $2$, this implies $|\sigma |+n$ is even, so terms involving fractional powers of $t$ vanish as claimed. This leads to exactly the sort of interior expansion described in Theorem 1.2.

2.8 The Heat Content on a Chart near the Boundary of $\Omega$

We now assume the coordinate chart meets the boundary. Again, we examine Eq. (2.10). We set $x=(r,y)$; the $d\nu _r$ integral ranges over $0\le r<\infty$ and the $d\nu _y$ integral ranges over $y\in \mathbb {R}^{m-1}$. The $dy$ and $d\tilde{y}$ integrals are handled using the analysis of Lemma 2.3 (1). We therefore suppress these variables and concentrate on the $d\nu _r$ integrals, and in essence assume that we are dealing with a one-dimensional problem; we can always choose the coordinates so $ds^2=dr^2+g_{ab}(r,y)dy^ady^b$. We resume the computation with Eq. (2.10), where we do not perform the integrals in the two variables normal to the boundary. We suppress other elements of the notation to examine an integral of the form

$$\begin{aligned} f(t):=t^{(n-1)/2} \int \limits _{x=0}^\infty \int \limits _{\tilde{x}=0}^\infty e^{-||x-\tilde{x}||_e^2/(4t)}(\textstyle \frac{X-\tilde{X}}{\sqrt{t}})^{\alpha _2} \langle r_{n,j,\alpha }(x,D_M)\phi (x),\rho (\tilde{x})\rangle d\nu _xd\nu _{\tilde{x}}. \end{aligned}$$

Here the integrand has compact support in $(x,\tilde{x})$ and is homogeneous of degree $n$ in the derivatives of the symbol of $D_M$, in the derivatives of $\phi$, and in the derivatives of $\rho$; there is no trouble with convergence. We suppress the role of $||\cdot ||_{g}$ in the tangential integrals which can also depend on the normal parameter. A crucial point is that the extra power of “$-\frac{1}{2}$” occurs in applying Lemma 2.3 to $\mathbb {R}^{m-1}$. We set

$$\begin{aligned}&f_1(t):=t^{(n-1)/2}\int \limits _{x=0}^\infty \int \limits _{\tilde{x} =-\infty }^0 e^{-||x-\tilde{x}||_e^2/(4t)}\\&\qquad \qquad \quad \times (\textstyle \frac{X-\tilde{X}}{\sqrt{t}})^{\alpha _2} \langle r_{n,j,\alpha }(x, D_M)\phi (x),\rho (\tilde{x})\rangle d\nu _xd\nu _{\tilde{x}}. \end{aligned}$$

Again, there is no trouble with convergence. The sum $f(t)+f_1(t)$ can then be handled as in Sect. 2.7 and gives rise to the interior term we have been studying. Thus, everything new comes from $f_1(t)$, and this is handled by Lemma 2.3 (2) with $\alpha =0$ after we replace $\tilde{x}$ by $-\tilde{x}$. The terms multiplying $t^{(n+1+|\alpha _1|+|\alpha _2|)/2}$ have degree $n+|\alpha _1|+|\alpha _2|$ in the derivatives of the symbol of $D_M$, of the derivatives of $\phi$, and of the derivatives of $\rho$. After setting $j=n+|\alpha _1|+|\alpha _2|$ and summing, we obtain the boundary terms of Theorem 1.2. We start out at $t^{(n-1)/2}$, but then we have two factors of $t^{1/2}$ arising from the $x$ and $\tilde{x}$ change of variable.

Remark 2.4

It is clear from the construction that the coefficients in the boundary asymptotic expansion depend holomorphically on the complex parameter $\alpha$ for $\mathfrak {R}(\alpha )<1$. It follows now that the constants $\varepsilon _{\nu ,\alpha }$ of Lemma 1.5 are holomorphic too.

3 The Proof of Theorem 1.6

We will establish Theorem 1.6 by evaluating the universal constants which appear in Lemma 1.5. In Sect. 3.1, we establish Lemma 1.4 which relates to the heat content asymptotics on the line. This result is then used in Sect. 3.2 to determine the constants $\{\varepsilon _{0,\alpha },\varepsilon _{1,\alpha },\varepsilon _{3,\alpha },\varepsilon _{4,\alpha }, \varepsilon _{7,\alpha },\varepsilon _{14,\alpha }\}$. Then in Sect. 3.3, we use product formulas to determine $\{\varepsilon _{6,\alpha },\varepsilon _{12,\alpha },\varepsilon _{13,\alpha }\}$. We complete the computation in Sect. 3.4 using warped products.

3.1 The Proof of Lemma 1.4

We apply the analysis of Sect. 2 to the one-dimensional setting. We work in the scalar setting and set $D=-\partial _x^2$. Consequently,

$$\begin{aligned} p_2(x,\xi )&= \xi ^2,\quad p_1(x,\xi )=0,\quad p_0(x)=0,\\ r_0(x,\xi ;\lambda )&= (\xi ^2-\lambda )^{-1},\quad r_n(x,\xi ;\lambda )=0\text { for }n\ge 1,\\ e_0(x,\xi ;t)&= e^{-t\xi ^2}\!,\quad e_n(x,\xi ;t)=0\text { for }n\ge 1. \end{aligned}$$

Consequently, we have $K_n=0$ for $n\ge 1$, while

$$\begin{aligned} K_0(x,\tilde{x};t)&= \frac{1}{2\pi }\int \limits _{-\infty }^\infty e^{-\sqrt{-1}(x-\tilde{x})\cdot \xi }e^{-t\xi ^2}d\xi \\&= \frac{1}{2\pi }e^{-(x-\tilde{x})^2/(4t)}\int \limits _{-\infty }^\infty e^{-t|\xi |^2}d\xi \\&= \frac{1}{\sqrt{4\pi t}}e^{-(x-\tilde{x})^2/(4t)}. \end{aligned}$$

This is not surprising, of course, as this is the heat kernel in flat space. Let

$$\begin{aligned}&f_1(t):=\frac{1}{\sqrt{4\pi t}}\int \limits _{x=0}^\infty \int \limits _{\tilde{x}=-\infty }^\infty e^{-(x-\tilde{x})^2/(4t)}\phi (x)\rho (\tilde{x})d\tilde{x} dx,\\&f_2(t):=\frac{1}{\sqrt{4\pi t}}\int \limits _0^\infty \int \limits _0^\infty e^{-(x+\tilde{x})^2/(4t)}\phi (x)\rho (-\tilde{x})d\tilde{x}dx. \end{aligned}$$

We may then express $\beta (\phi ,\rho ,D_M)(t)=f_1(t)-f_2(t)$. We change variables setting $\tilde{x}=x+u$ to express

$$\begin{aligned} f_1(t)\sim \frac{1}{\sqrt{4\pi t}}\int \limits _{x=0}^\infty \int \limits _{u=-\infty }^\infty \sum _{k=0}^\infty \frac{1}{k!}\phi (x)\rho ^{(k)}(x)u^k e^{-u^2/(4t)}dudx. \end{aligned}$$

In the sum, we must have $k=2\bar{k}$ is even. We integrate by parts $\bar{k}$ times to evaluate the constants which appear. Alternatively, we change variables $u^2=4tv$ and use standard formulae for the $\Gamma$-function to obtain that

$$\begin{aligned}&\frac{1}{\sqrt{4\pi t}}\int \limits _{-\infty }^\infty \frac{1}{(2\bar{k})!}u^{2\bar{k}}e^{-u^2/(4t)}du =\frac{2}{\sqrt{4\pi t}(2\bar{k})!}\int \limits _0^\infty u^{2\bar{k}}e^{-u^2/(4t)}du\\&\quad =\frac{2^{2\bar{k}}t^{\bar{k}}}{\sqrt{\pi }(2\bar{k})!}\int \limits _0^{\infty }v^{\bar{k}-\frac{1}{2}}e^{-v}dv =\frac{2^{2\bar{k}}\Gamma ( \bar{k}+\frac{1}{2})t^{\bar{k}}}{\sqrt{\pi }(2\bar{k})!}=\frac{t^{\bar{k}}}{\bar{k}!}. \end{aligned}$$

The interior terms arise from expanding

$$\begin{aligned} f_1(t)\sim \sum _{\bar{k}}\frac{t^{\bar{k}}}{\bar{k}!}\int \limits _{0}^\infty \phi (x)\rho ^{(2\bar{k})}(x)dx \sim \sum _{\bar{k}}(-1)^{\bar{k}}\frac{t^{\bar{k}}}{\bar{k}!}\int \limits _0^\infty \phi (x)D^{\bar{k}}\rho (x)dx. \end{aligned}$$

Next we evaluate $f_2$ (and we have to subtract this term). We expand

$$\begin{aligned} \phi (x)\sim x^{-\alpha }\sum _i\phi _ix^i\text { and }\rho (\tilde{x})\sim \sum _j\rho _j\tilde{x}^j. \end{aligned}$$

We do not put in the factorials, so $\rho _j=\frac{1}{j!}\rho ^{(j)}(0)$.

$$\begin{aligned} f_2(t)\sim \frac{1}{\sqrt{4\pi t}}\int \limits _{0}^\infty \int \limits _0^\infty e^{-(x+\tilde{x})^2/(4t)} \sum _{i,j} \phi _i\rho _jx^{i-\alpha }\tilde{x}^jd\nu _xd\nu _{\tilde{x}} . \end{aligned}$$

We change variables to set $x=\sqrt{t} u$ and $\tilde{x}=\sqrt{t}\tilde{u}$ to complete the proof of Lemma 1.4 by expressing

$$\begin{aligned} f_2(t)\sim \frac{1}{\sqrt{4\pi }}\sum _{i,j}t^{(i+j-\alpha +1)/2}(-1)^j\phi _i\rho _j \times \int \limits _{0}^\infty \int \limits _{0}^\infty e^{-(u+\tilde{u})^2/4}u^{i-\alpha }\tilde{u}^jdud\tilde{u}. \square \end{aligned}$$

3.2 Evaluating the Constants for the One-Dimensional Case

We use, e.g., Mathematica [14] to compute the coefficients of Lemma 1.5:

$$\begin{aligned} \varepsilon _{0,\alpha }&= -\frac{1}{\sqrt{4\pi }}\int \limits _0^\infty \int \limits _0^\infty x^{-\alpha }e^{-(x+y)^2/4} dxdy=\frac{2^{1-\alpha }\Gamma (\frac{2-\alpha }{2})}{(-1+\alpha )\sqrt{4\pi }}=\frac{c_\alpha }{2} \\ \varepsilon _{1,\alpha }&= -\frac{1}{\sqrt{4\pi }}\int \limits _0^\infty \int \limits _0^\infty x^{1-\alpha }e^{-(x+y)^2/4} dxdy=\frac{2^{2-\alpha }\Gamma (\frac{3-\alpha }{2})}{(-2+\alpha )\sqrt{4\pi }}=\frac{c_{\alpha -1}}{2} \\ \varepsilon _{3,\alpha }&= +\frac{1}{\sqrt{4\pi }}\int \limits _0^\infty \int \limits _0^\infty x^{-\alpha }ye^{-(x+y)^2/4} dxdy=-\frac{2^{1-a}\Gamma (\frac{1-\alpha }{2})}{(-2+\alpha )\sqrt{4\pi }}=\frac{-c_{\alpha -1}}{2(1-\alpha )} \\ \varepsilon _{4,\alpha }&= -\frac{1}{\sqrt{4\pi }}\int \limits _0^\infty \int \limits _0^\infty x^{2-\alpha }e^{-(x+y)^2/4} dxdy=\frac{2^{3-\alpha }\Gamma (\frac{4-\alpha }{2})}{(-3+\alpha )\sqrt{4\pi }}=\frac{c_{\alpha -2}}{2} \\ \varepsilon _{14,\alpha }&= +\frac{1}{\sqrt{4\pi }}\int \limits _0^\infty \int \limits _0^\infty x^{1-\alpha }ye^{-(x+y)^2/4} dxdy=-\frac{2^{2-a}\Gamma (\frac{2-\alpha }{2})}{(-3+\alpha )\sqrt{4\pi }}=\frac{-c_{\alpha -2}}{2(2-\alpha )} \\ \varepsilon _{7,\alpha }&= -\frac{1}{\sqrt{4\pi }}\int \limits _0^\infty \int \limits _0^\infty x^{-\alpha }y^2e^{-(x+y)^2/4} dxdy=-\frac{2^{3-\alpha }\Gamma (\frac{2-\alpha }{2})}{(3-4\alpha +\alpha ^2)\sqrt{4\pi }} \\&= \frac{c_{\alpha -2}}{(2-3\alpha +\alpha ^2)}. \end{aligned}$$

3.3 Product Formulae

We evaluate $\{\varepsilon _{6,\alpha },\varepsilon _{12,\alpha },\varepsilon _{13,\alpha }\}$. Let $(N,g_N)$ be a closed Riemannian manifold, let $D_N$ be the scalar Laplacian on $N$, let $S=(S^1,dx^2)$, and let $D_S=-\partial _x^2$, where $x$ is the usual periodic angular parameter on the circle $S^1$. We consider the product manifold. Let

$$\begin{aligned} \begin{array}{ll} (M,g_M):=(N\times S^1,g_N+dx^2),&{}D_M:=D_N+D_S,\\ \Sigma :=[0,\pi ],&{}\Omega =N\times \Sigma ,\\ \phi (x,z)=\phi _\Sigma (x)\phi _N(z),&{}\rho (x,z)=\rho _\Sigma (x)\rho _N(z) \end{array} \end{aligned}$$

for $\phi _N\in C^\infty (N)$, $\rho _N\in C^\infty (N)$, $\phi _\Sigma \in \mathcal {K}_\alpha (\Sigma )$, and $\rho _\Sigma \in C^\infty (\Sigma )$. As the structures decouple,

$$\begin{aligned}&e^{-tD_M}=e^{-tD_S}e^{-tD_N}\text { so}\\&\beta _\Omega (\phi ,\rho ,D_M)(t)=\beta _\Sigma (\phi ,\rho ,D_S)(t)\cdot \beta _N(\phi _N,\rho _N,D_N)(t). \end{aligned}$$

There are, of course, no boundary terms in the asymptotic series for $\beta _N$, and the interior terms are given by Lemma 1.4. Equating the two asymptotic series then permits us to conclude that $\beta _{2,\alpha }^{\partial \Omega }(\phi ,\rho ,D_M)=\mathcal {E}_2+\mathcal {E}_0$, where

$$\begin{aligned}&\mathcal {E}_2:=\int \limits _{\partial \Sigma }\beta _{2,\alpha }^{\partial \Sigma } (\phi _\Sigma ,\rho _\Sigma ,D_\Sigma )(x)dx\cdot \int \limits _N\phi _N(z)\rho _N(z)dz\text { and}\\&\mathcal {E}_0:=\frac{1}{2}c_{\alpha -2}\int \limits _\Sigma \phi _{\Sigma ,0}(y)\rho _{\Sigma ,0}(y)dy\cdot \int \limits _N-\phi _N(z)\cdot D_N\rho _N(z)dz. \end{aligned}$$

We suppress terms not of interest to equate

$$\begin{aligned}&\int \limits _{\partial \Sigma }\int \limits _N\phi _{\Sigma ,0}(x)\rho _{\Sigma ,0}(x)\\&\qquad \cdot \left\{ \varepsilon _{6,\alpha }E_N(z)\phi _N(z)\rho _N(z)+\varepsilon _{12,\alpha }\phi _{N:a}\rho _{N:a} +\varepsilon _{13,\alpha }\tau \phi _N\rho _N\right\} dxdz\\&= \frac{1}{2}c_\alpha \int \limits _{\partial \Sigma }\phi _{\Sigma ,0}(x)\rho _{\Sigma ,0}(x)dx\cdot \int \limits _N\phi _N(z)(\rho _{N;aa}+E)\rho _N(z)dz. \end{aligned}$$

Integrating by parts on $N$ permits us to see

$$\begin{aligned} \int \limits _N\phi _N(z)\rho _{N;aa}(z)dz=-\int \limits _N\phi _{N;a}(z)\rho _{N;a}(z)dz. \end{aligned}$$

Applying the recursion relation of Eq. (1.6) then yields

$$\begin{aligned} -\varepsilon _{12,\alpha }=\varepsilon _{6,\alpha }=\frac{1}{2}c_\alpha =-\frac{\alpha -3}{2(\alpha -1)(\alpha -2)}\frac{1}{2}c_{\alpha -2},\quad \text { and } \quad \varepsilon _{13,\alpha }=0. \end{aligned}$$

3.4 Warped Products

We now determine the coefficients involving $L$ and $\mathrm{Ric }$. The fact that we are working with quite general operators is now crucial. We extend the formalism of Sect. 3.3 and consider warped products. Let

$$\begin{aligned} \Sigma =[0,\pi ],\quad S=[0,2\pi ]/0\sim 2\pi ,\quad D_S=-\partial _x^2. \end{aligned}$$

Let $\mathbb {T}^{m-1}$ be the torus with periodic parameters $(\theta _1,\ldots ,\theta _{m-1})$, let $M= \mathbb {T}^{m-1}\times S^1$, and let $\Omega = \mathbb {T}^{m-1}\times \Sigma$. Let $f_a\in C^\infty (S)$ be a collection of smooth functions satisfying $f_a(0)=0$ and $f_a\equiv 0$ near $x=\pi$. Let $\delta _a\in \mathbb {R}$. Set

$$\begin{aligned} \begin{array}{ll} M:=\mathbb {T}^{m-1}\times S,&{} \displaystyle ds^2_M=\sum _{a=1}^{m-1}e^{2f_a(x)}d\theta _a\circ d\theta _a+dx\circ dx,\\ \Omega :=\mathbb {T}^{m-1}\times [0,\pi ],&{}\displaystyle D_M:=-\sum _{a=1}^{m-1}e^{-2f_a(x)}(\partial _{\theta _a}^2+\delta _a\partial _{\theta _a})-\partial _x^2. \end{array} \end{aligned}$$

Let $\phi _\Sigma \in \mathfrak {K}_\alpha (\Sigma )$ with $\phi _\Sigma$ vanishing identically near $\pi$, and let $\rho _\Sigma \in C^\infty ([0,\pi ])$ with $\rho _\Sigma$ vanishing identically near $\pi$ as well. Set

$$\begin{aligned} \phi _\Omega (x,y)=\phi _\Sigma (x)\text { and }\rho _\Omega (x,y)=\rho _\Sigma (x)e^{-\sum _if_a(x)}. \end{aligned}$$

Lemma 3.1

$\beta _{j,\alpha }^{\partial M}(\phi _\Omega ,\rho _\Omega ,D_M) =\beta _{j,\alpha }^{\partial \Sigma }(\phi _\Sigma ,\rho _\Sigma ,D_S)\mathrm{vol }(\mathbb {T}^{m-1})$ for $j\ge 0$.

Proof

Note that $x$ is the geodesic distance to $\{0\}$ in $\Sigma$ and that $x$ is the geodesic distance to $\{0\times \mathbb {T}^{m-1}\}$ in $\Omega$; the component where $x=\pi$ plays no role as $\phi$ and $\rho$ vanish identically near this component. Since $\phi _\Omega$ is independent of $y\in \mathbb {T}^{m-1}$, the problem decouples and

$$\begin{aligned} \left\{ e^{-tD_M}\phi _\Omega \right\} (x,y;t)=\left\{ e^{-tD_S}\phi _\Sigma \right\} (x;t). \end{aligned}$$

The Riemannian measure on $M$ takes the form

$$\begin{aligned} d\nu _M=\sqrt{\det g_{ij}}dydx=e^{\sum _af_a}dydx. \end{aligned}$$

Since $\rho _\Omega d\nu _\Omega =\rho _\Sigma dxdy$, we have

$$\begin{aligned} \beta _{\Omega }(\phi _\Omega ,\rho _\Omega ,D_M)(t)=\beta _\Sigma (\phi _\Sigma ,\rho _\Sigma ,D_S)(t)\cdot \mathrm{vol }(\mathbb {T}^{m-1}). \end{aligned}$$

Lemma 3.1 now follows for $\alpha \not \in \mathbb {Z}$, since the interior invariants and the boundary invariants do not interact. Since the invariants $\beta _{j,\alpha }^{\partial \Omega }$ and $\beta _{j,\alpha }^{\partial \Sigma }$ are analytic in $\alpha$, the desired conclusion also follows for $\alpha \in \mathbb {Z}$. Thus, even if one were only interested in the case $\alpha =0$, it is convenient to have more general values of $\alpha$ available. $\square$

We apply Lemma 1.5. Although the structures are flat on $\Sigma$, they are not flat on $\Omega$, and this makes all the difference. We determine the relevant tensors as follows:

$$\begin{aligned} \begin{array}{ll} \Gamma _{abm}=-f_a^\prime \delta _{ab}e^{2f_a},&{}\Gamma _{ab}^m=-f_a^\prime e^{2f_a}\delta _{ab},\\ \Gamma _{amb}=f_a^\prime \delta _{ab}e^{2f_a} ,&{}\Gamma _{am}^b=f_a^\prime \delta _{a,b}, \\ L_{ab}=\Gamma _{ab}^m|_{\partial M}=-f_a^\prime \delta _{ab},&{} \\ \textstyle \omega _a={\textstyle \frac{1}{2}}e^{2f_a}\delta _a,&{}\tilde{\omega }_a=-\omega _a =-{\textstyle \frac{1}{2}}e^{2f_a}\delta _a, \\ \omega _m=-{\textstyle \frac{1}{2}}{\sum \nolimits _{a}}f_a^\prime ,&{} \tilde{\omega }_m=-\omega _m={\textstyle \frac{1}{2}}{{\sum }_{a}}{f_{a}^{\prime }}. \end{array} \end{aligned}$$

Consequently,

$$\begin{aligned} R_{ambm}&= g((\nabla _a\nabla _m-\nabla _m\nabla _a)e_b,e_m) =\Gamma _{ac}^m\Gamma _{mb}^c-\partial _m\Gamma _{ab}^m \\&= \{-(f_a^\prime )^2+f_a^{\prime \prime }+2(f_a^\prime )^2\}e^{2f_a}\delta _{ab}, \\ \mathrm{Ric }_{mm}&= -\textstyle \sum _a\left\{ f_a^{\prime \prime }+(f_a^\prime )^2\right\} , \\ E|_{\partial M}&= -\partial _m\omega _m-\omega _a^2-\omega _m^2+\omega _m\Gamma _{aa}^m \\&= \textstyle \frac{1}{2}\sum _af_a^{\prime \prime }-\frac{1}{4}\sum _a\delta _a^2-\frac{1}{4}\sum _{a,b}f_a^\prime f_b^\prime +\frac{1}{2}\sum _{a,b}f_a^\prime f_b^\prime \\&= \textstyle \frac{1}{2}\sum _af_a^{\prime \prime }-\frac{1}{4}\sum _a\delta _a^2+\frac{1}{4}\sum \limits _{a,b}f_a^\prime f_b^\prime . \end{aligned}$$

We introduce the notation $\phi _\Omega$ and $\rho _\Omega$ to emphasize that we are computing with the structures on $M$ and not on $S$. We evaluate on the component $x=0$:

$$\begin{aligned} \phi _{\Omega ,0}|_{x=0}&= \phi _{\Sigma ,0}(0), \\ \phi _{\Omega ,1}|_{x=0}&= \{\nabla _{\partial x}(\phi _{\Sigma ,0}+x\phi _{\Sigma ,1})\}|_{x=0} =\{(\partial _x-\textstyle {\textstyle \frac{1}{2}}{{\sum }_{a}}{f_{a}^{\prime }})(\phi _{\Sigma ,0}+x\phi _{\Sigma ,1})\}|_{x=0} \\&= -{\textstyle \frac{1}{2}}{{\sum }_{a}}{f_{a}^{\prime }}\phi _{\Sigma ,0}(0)+\phi _{\Sigma ,1}(0), \\ \textstyle \phi _{\Omega ,2}|_{x=0}&= {\textstyle \frac{1}{2}}\{(\nabla _{\partial _x})^2 (\phi _{\Sigma ,0}+x\phi _{\Sigma ,1}+x^2\phi _{\Sigma ,2})\}|_{x=0} \\&= {\textstyle \frac{1}{2}}\{(\partial _x -\textstyle {\textstyle \frac{1}{2}}{{\sum }_{a}}{f_{a}^{\prime }})^2(\phi _{\Sigma ,0}+x\phi _{\Sigma ,1}+x^2\phi _{\Sigma ,2})\}|_{\partial \Sigma } \\&= \{{\textstyle \frac{1}{8}}{{\sum }_{a,b}}{f_{a}^{\prime }} {f_{b}^{\prime }} -{\textstyle \frac{1}{4}}{{\sum }_{a}}{f_{a}^{\prime \prime }}\}\phi _{\Sigma ,0}(0)-\frac{1}{2}{{\sum }_{a}}{f_{a}^{\prime }}\phi _{\Sigma ,1}(0) +\phi _{\Sigma ,2}(0), \\ \rho _{\Omega ,0}|_{x=0}&= \rho _{\Sigma ,0}(0), \\ \rho _{\Omega ,1}|_{x=0}&= \{\tilde{\nabla }_{\partial x}(\rho )\}|_{x=0} =\{(\partial _x+{\textstyle \frac{1}{2}}{{\sum }_{a}}{f_{a}^{\prime }})(e^{-{{\sum }_{a}}{f_{a}}}) (\rho _{\Sigma ,0}+x\rho _{\Sigma _1})\}|_{x=0} \\&= -{\textstyle \frac{1}{2}}{{\sum }_{a}}{f_{a}^{\prime }}\rho _{\Sigma ,0}(0)+\rho _{\Sigma ,1}(0), \\ \rho _{\Omega ,2}|_{x=0}&= {\textstyle \frac{1}{2}}\{(\tilde{\nabla }_{\partial r})^2\rho _\Sigma \}|_{x=0} ={\textstyle \frac{1}{2}}\{(\partial _x +{\textstyle \frac{1}{2}}{{\sum }_{a}}{f_{a}^{\prime }})^2(e^{-{{\sum }_{a}}{f_{a}}})\}|_{x=0} \\&= \{\textstyle {\textstyle \frac{1}{8}}{\sum _{.a,b}}{f_{a}^{\prime }} {f_{b}^{\prime }} -{\textstyle \frac{1}{4}}{{\sum }_{a}}{f_{a}^{\prime \prime }}\}\rho _{\Sigma ,0}(0) -{\textstyle \frac{1}{2}}{{\sum }_{a}}{f_{a}^{\prime }}\rho _{\Sigma ,1}(0)+\rho _{\Sigma ,2}(0), \\ \phi _{\Omega ,0:a}\rho _{\Omega ,0:a}&= -\frac{1}{4}{\sum \nolimits _{a}}\delta _a^2. \end{aligned}$$

The structures defined by $f_a^\prime$, $f_a^{\prime \prime }$, and $\delta _a$ do not appear in $\beta _{j,\alpha }^{\partial \Sigma }$, and thus these terms must give zero in $\beta _{j,\alpha }^{\partial \Omega }$. By considering the monomial ${\sum _{a,b}}f_a^\prime f_b^\prime \phi _{\Sigma ,0}(0)\rho _{\Sigma ,0}(0)$ in $\beta _{2,\alpha }^{\partial M}$, we obtain the relation

$$\begin{aligned} \textstyle \frac{1}{8}\varepsilon _{4,\alpha } +{\textstyle \frac{1}{2}}\varepsilon _{5,\alpha }+{\textstyle \frac{1}{4}}\varepsilon _{6,\alpha } +{\textstyle \frac{1}{8}}\varepsilon _{7,\alpha } +{\textstyle \frac{1}{2}}\varepsilon _{8,\alpha } +\varepsilon _{10,\alpha }{+\frac{1}{4}\varepsilon _{14,\alpha }}=0. \end{aligned}$$

We obtain other relations by considering suitable monomials:

$$\begin{aligned} \begin{array}{ll} \hbox {Relation}&{}\hbox {Monomial}\\ -{\textstyle \frac{1}{2}}\varepsilon _{1,\alpha }-\varepsilon _{2,\alpha }-{{\textstyle \frac{1}{2}}\varepsilon _{3,\alpha }}=0, &{}{\sum \nolimits _{a}}f_a^\prime \phi _{\Sigma ,0}(0)\rho _{\Sigma ,0}(0),\\ -{\textstyle \frac{1}{4}}(\varepsilon _{6,\alpha }+\varepsilon _{12,\alpha })=0,&{} {\sum \nolimits _{a}}\delta _a^2\phi _{\Sigma ,0}(0)\rho _{\Sigma ,0}(0),\\ \textstyle -\varepsilon _{9,\alpha }+\varepsilon _{11,\alpha }=0,&{} {\sum \nolimits _{a}}(f_a^\prime )^2\phi _{\Sigma ,0}(0)\rho _{\Sigma ,0}(0),\\ -{\textstyle \frac{1}{2}}\varepsilon _{4,\alpha }-\varepsilon _{5,\alpha }-{{\textstyle \frac{1}{2}}\varepsilon _{14,\alpha }}=0,&{} {\sum \nolimits _{a}}f_a^\prime \phi _{\Sigma ,1}(0)\rho _{\Sigma ,0}(0),\\ -{\textstyle \frac{1}{2}}\varepsilon _{14,\alpha }-\varepsilon _{8,\alpha }-{{\textstyle \frac{1}{2}}\varepsilon _{7,\alpha }}=0,&{} {\sum \nolimits _{a}}f_a^\prime \phi _{\Sigma ,0}(0)\rho _{\Sigma ,1}(0),\\ -{\textstyle \frac{1}{4}}\varepsilon _{4,\alpha } +{\textstyle \frac{1}{2}}\varepsilon _{6,\alpha }-{\textstyle \frac{1}{4}}\varepsilon _{7,\alpha } -\varepsilon _{9,\alpha }=0,&{} {\sum \nolimits _{a}}f_a^{\prime \prime }\phi _{\Sigma ,0}(0)\rho _{\Sigma ,0}(0).\end{array} \end{aligned}$$

Theorem 1.6 follows from these equations and the relations established previously.

4 Further Functorial Properties

In Sect. 4.1, we examine dimension shifting, and in Sect. 4.2, we relate the Dirichlet and Neumann heat content asymptotics to the asymptotics we have been studying. In addition to providing useful crosschecks on our work, these properties are worth noting, as they promise to be useful in other contexts.

4.1 Dimension Shifting

Let $\mathfrak {R}(\alpha )<<0$. If $\phi \in \mathcal {K}_{\alpha -1}$, then we may regard $\phi$ as defining an element $\tilde{\phi }\in \mathcal {K}_{\alpha }$. If we expand $\phi =r^{-\alpha +1}(\phi _0+r\phi _1+\cdots )$, then $\tilde{\phi }=r^{-\alpha }(0+r\phi _0+r^2\phi _1+\cdots )$. Consequently, $\tilde{\phi }_i=\phi _{i-1}$ for $i\ge 1$, and

$$\begin{aligned} \beta _{j,\alpha }^{\partial \Omega }(\tilde{\phi },\tilde{\rho },D_M) =\beta _{j-1,\alpha -1}^{\partial \Omega }(\phi ,\rho ,D_M). \end{aligned}$$

Examining the formulas of Lemma 1.5 then yields the relations

$$\begin{aligned} \begin{array}{llll} \varepsilon _{1,\alpha }=\varepsilon _{0,\alpha -1},&\varepsilon _{4,\alpha }=\varepsilon _{0,\alpha -2},&\varepsilon _{5,\alpha }=\varepsilon _{2,\alpha -1},&\varepsilon _{14,\alpha }=\varepsilon _{3,\alpha -1}. \end{array} \end{aligned}$$

These hold, of course, only if $\mathfrak {R}(\alpha )<0$; we use analytic continuation to derive the general result. Once again, it is convenient to have values of $\alpha$ other than $\alpha =0$.

4.2 The Dirichlet and Neumann Heat Content Asymptotics

There is a useful relationship between the heat content asymptotics being studied at present and the ones studied previously. Let $\beta _D$ (resp., $\beta _N$) be the heat content corresponding to Dirichlet (resp., Neumann) boundary conditions.

Lemma 4.1

Let $(M,g)$ be a closed Riemannian manifold. Let $T$ be an isometric involution of $M$ with $\mathrm{Fix }(T)=N$ a totally geodesic submanifold of codimension $1$. Assume $M-N$ decomposes as the union of two open submanifolds $M_+\cup M_-$ which are interchanged by $T$. Let $\Omega _\pm :=M_\pm \cup N$. Then

$$\begin{aligned} \beta _\Omega (\phi ,\rho ,\Delta )(t)= \textstyle \frac{1}{2}\{\beta _D(\phi ,\rho ,\Delta )(t)+\beta _N(\phi ,\rho ,\Delta )(t)\}. \end{aligned}$$

Proof

We can use the $\mathbb {Z}_2$ involution $T$ to choose a spectral resolution

$$\begin{aligned} \{\lambda _N,\phi _{N,n}\}_{n=1}^\infty \cup \{\lambda _D,\phi _{D,n}\}_{n=1}^\infty \end{aligned}$$

for $L^2(M)$ so $T^*\phi _{N,n}=-\Phi _{N,n}$ and $T^*\phi _{D,n}=\Phi _{D,n}$. Then $\{\lambda _{N,n},\sqrt{2}\Phi _{N,n}\}_{n=1}^\infty$ is a spectral resolution for the Neumann Laplacian on $\Omega _+$ and $\{\lambda _{D,n},\sqrt{2}\Phi _{D,n}\}_{n=1}^\infty$ is a spectral resolution for the Dirichlet Laplacian on $\Omega _+$. We compute that

$$\begin{aligned} e^{-t\Delta _D}\phi&= 2{\sum _{n}}e^{-t\lambda _{N,n}}(\phi ,\phi _{N,n})_{L^2}\phi _{N,n},\\ e^{-t\Delta _N}\phi&= 2{\sum _{n}}e^{-t\lambda _{D,n}}(\phi ,\phi _{D,n})_{L^2}\phi _{D,n},\\ e^{-t\Delta }\phi&= {\sum _{n}}e^{-t\lambda _{N,n}}(\phi ,\phi _{N,n})_{L^2}\phi _{N,n} +{\sum _{n}}e^{-t\lambda _{D,n}}(\phi ,\phi _{D,n})_{L^2}\phi _{D,n},\\&= \frac{1}{2}\{e^{-t\Delta _D}\phi +e^{-t\Delta _N}\phi \}. \end{aligned}$$

The desired result now follows by taking the inner product with $\rho$ and integrating over the support of $\rho$ which is $\Omega _+$. $\square$

This is not useful for studying the terms which involve the second fundamental form $L_{ab}$. However, if we average the remaining terms for Dirichlet and Neumann boundary conditions in Theorem 1.8, we get the analogous terms in Theorem 1.6.

5 Reduction to the Closed Setting

In Sect. 2, we used the classic calculus of pseudo-differential operators depending on a complex parameter which was developed by Seeley [12, 13]. That formalism is valid only for compact and closed manifolds. In this section, we will derive Theorem 1.3 where $(M,g)=(\mathbb {R}^m,g_e)$ or where $(M,g)$ is a compact subset of $\mathbb {R}^m$ of dimension $m$ from Theorem 1.2. The latter dealt with compact manifolds without boundary.

We adopt the following notational conventions. Let $\Omega \subset \mathrm{Interior }\{\tilde{M}\}\subset \tilde{M}\subset M$, where $\Omega$ and $\tilde{M}$ are compact manifolds of dimension $m$ with smooth boundaries. Let $\epsilon :=dist_g(\partial \Omega ,\partial \tilde{M})>0$. Let $\beta _\Omega ^{\tilde{M}}$ be the heat content of $\Omega$ in $\tilde{M}$ and let $\beta _\Omega ^M$ be the heat content of $\Omega$ in $M$.

Theorem 5.1

Assume that $(M,g)$ is complete with non-negative Ricci curvature. Let $\rho$ be continuous on $M$ and let $\phi \in L^1(\Omega )$. Then,

$$\begin{aligned} |\beta _\Omega ^{M}(\phi ,\rho ,\Delta _g)(t)- \beta _\Omega ^{\tilde{M}}(\phi ,\rho ,\Delta _g)(t)|\le 2^{(2+m)/2} ||\phi ||_{L^1(\Omega )}||\rho ||_{L^{\infty }(\Omega )}e^{-\epsilon ^2/(8t)}. \end{aligned}$$

Proof

Let $\tilde{K}$ be the Dirichlet heat kernel for $\tilde{M}$. By minimality,

$$\begin{aligned} 0\le \tilde{K}(x,\tilde{x};t)\le K(x,\tilde{x};t) \end{aligned}$$

for all $x\in \tilde{M},\tilde{x}\in \tilde{M}, t>0$. Since $M$ is stochastically complete, we have that

$$\begin{aligned} 1= \int \limits _MK(x,\tilde{x};t)d\tilde{x}. \end{aligned}$$

Moreover, since $M$ and hence $\tilde{M}$ have non-negative Ricci curvature, we have by Theorem 3.5.3 in [7] (see also Lemma 5 in [2]) that

$$\begin{aligned} \int \limits _{\tilde{M}}\tilde{K}(x,\tilde{x};t)d\tilde{x}&\ge 1-2^{(2+m)/2}e^{-dist _g(x,\partial \tilde{M})^2/(8t)}\nonumber \\&=\int \limits _M K(x,\tilde{x};t)d\tilde{x}-2^{(2+m)/2}e^{-dist _g(x,\partial \tilde{M})^2/(8t)}\nonumber \\&\ge \int \limits _M K(x,\tilde{x};t)d\tilde{x}-2^{(2+m)/2}e^{-\epsilon ^2/(8t)},\ \ x\in \Omega ,\ t>0. \end{aligned}$$

(5.1)

So

$$\begin{aligned}&|\beta _\Omega (\phi ,\rho ,\Delta _g)(t)-\tilde{\beta }_\Omega (\phi ,\rho ,\Delta _g)(t)| \\&\quad =\left| \int \limits _{\Omega }\int \limits _{\Omega }(K(x,\tilde{x};t)-\tilde{K}(x,\tilde{x};t))\phi (x)\rho (\tilde{x})dxd\tilde{x}\right| \\&\quad \le ||\rho ||_{L^{\infty }(\Omega )}\int \limits _{\Omega }\int \limits _{\Omega }(K(x,\tilde{x};t)-\tilde{K}(x,\tilde{x};t))|\phi (x)|dxd\tilde{x} \\&\quad \le ||\rho ||_{L^{\infty }(\Omega )}\int \limits _{\Omega }|\phi (x)|dx\left( \int \limits _{\tilde{M}}(K(x,\tilde{x};t)-\tilde{K}(x,\tilde{x};t))d\tilde{x}\right) \\&\quad \le ||\rho ||_{L^{\infty }(\Omega )}\int \limits _{\Omega }|\phi (x)|dx\left( \int \limits _{M}K(x,\tilde{x};t)d\tilde{x}-\int \limits _{\tilde{M}}\tilde{K}(x,\tilde{x};t)d\tilde{x}\right) . \end{aligned}$$

Theorem 5.1 now follows by Eq. (5.1). $\square$

Suppose that $\mathcal {N}=(\mathbb {R}^m,g_e)$ or that $\mathcal {N}$ is a compact subset of dimension $m$ with smooth boundary in $(\mathbb {R}^m,g_e)$. The following lemma will permit us to deduce Theorem 1.2 for $\mathcal {N}$ from the corresponding assertion for closed ambient manifolds by using Theorem 5.1 to localize matters to a small neighborhood $\tilde{M}$ of $\Omega$.

Lemma 5.2

Let $(\tilde{M},g_e)$ be a compact smooth manifold of dimension $m$ which is contained in $\mathbb {R}^m$. Then $(\tilde{M},g_e)$ is isometric to a compact smooth manifold of a flat $m$-dimensional torus.

Proof

Let $B_r(0)$ be the ball of radius $r$ about the origin in $\mathbb {R}^m$. Since $\tilde{M}$ is compact, $\tilde{M}$ is contained in $B_n(0)$ for some positive integer $n$. Let $\Gamma :=\{2n\mathbb {Z}\}^{m}$ be the rescaled integer lattice and let $\mathcal {T}^m=\mathbb {R}^m/\Gamma$ be a flat torus. Then $\tilde{M}\subset B_n(0)$ embeds isometrically in $\mathcal {T}^m$. $\square$