Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 61–64

Heat Content Asymptotics of Riemannian Manifolds

M. van den Berg, S. Desjardins, and P. B. Gilkey

Let M be a compact Riemannian manifold with smooth boundary ∂M, let V be a smooth vector bundle over M, let ∇ be a connection on V, and let E ∈ C^{∞}(End(V)) be an auxiliary endomorphism of V. Let

D = −(g^{ij}∇_{i}∇_{j} + E)

be an operator of Laplace type on V. Let S ∈ C^{∞}(End(V)) and let ∇_{m} be covariant differentiation with respect to the inward pointing unit normal. Let B_{S}^{+}f = (∇_{m}^{D}f + Sf)|_{∂M} and B^{−}f = f|_{∂M} be Neumann and Dirichlet boundary conditions. Let h(x, t) = e^{−tD_{B}}f_{1}(x) for B ∈ {B_{S}^{+}, B^{−}} be the solution for t > 0 to the equations:

∂_{t}h(x, t) = −Dh(x, t), Bh(x, t) = 0, and lim_{t→0^{+}} h(x, t) = f_{1}(x).

Denote the natural pairing between V and the dual bundle V^{∗} by ⟨·, ·⟩. Let dx and dy be the Riemannian measures on M and ∂M. To study the short time behavior of h, introduce an auxiliary smooth test function f_{2} ∈ C^{∞}(V^{∗}) and define:

\begin{matrix}β(f_{1}, f_{2}, D, B)(t) = & ∫_{M}⟨e^{−tD_{B}}f_{1}, f_{2}⟩dx \\ ≃ & Σ_{n=0}^{∞}β_{n}(f_{1}, f_{2}, D, B)t^{n/2}. \end{matrix}

There exist local invariants β_{n}^{int}(f_{1}, f_{2}, D) and β_{n}^{bd}(f_{1}, f_{2}, D, B) so

β_{n}(f_{1}, f_{2}, D, B) = ∫_{M}β_{n}^{int}(f_{1}, f_{2}, D)dx + ∫_{∂M}β_{n}^{bd}(f_{1}, f_{2}, D, B)dy.

Let indices a, b range from 1 through m − 1 and index a local orthonormal frame for the tangent bundle of the boundary; adopt the Einstein convention and sum over repeated indices. Let R_{ijkl} be the curvature tensor of the Levi-Civita connection ∇^{M} of M; with our sign convention R_{1212} = −1 for the standard sphere in R^{3}. Let τ = R_{ijji} and ρ_{ij} = R_{ikkj} be the scalar curvature and the Ricci tensor. Let “;” and “:” be multiple covariant differentiation with respect to ∇^{M} and ∇^{∂M}. When sections of bundles built from V are involved, “;” and “:” will mean

∇^{M} ⊗ 1 + 1 ⊗ ∇^{D} and ∇^{∂M} ⊗ 1 + 1 ⊗ ∇^{D}.

62 M. van den Berg, S. Desjardins, P.B. Gilkey

Let L_{ab} = (e_{m}, ∇_{e_{a}}^{M}e_{b}) be the second fundamental form and let D̃ be the formal adjoint on C^{∞}(V^{∗}); let B̃ be the adjoint boundary condition.

Theorem 1: (see [3, 5])

(a) β_{0}(f_{1}, f_{2}, D, B^{−}) = ∫_{M}⟨f_{1}, f_{2}⟩dx.

(b) β_{1}(f_{1}, f_{2}, D, B^{−}) = −2π^{−1/2}∫_{∂M}⟨f_{1}, f_{2}⟩dy.

(c) β_{2}(f_{1}, f_{2}, D, B^{−}) = −∫_{M}⟨Df_{1}, f_{2}⟩dx + ∫_{∂M}\{⟨\frac{1}{2}L_{aa}f_{1}, f_{2}⟩ − ⟨f_{1}, f_{2;m}⟩\}dy.

(d) β_{3}(f_{1}, f_{2}, D, B^{−}) = −2π^{−1/2}∫_{∂M}\{\frac{2}{3}⟨f_{1;mm}, f_{2}⟩ + \frac{2}{3}⟨f_{1}, f_{2;mm}⟩ − ⟨f_{1:a}, f_{2:a}⟩ +⟨Ef_{1}, f_{2}⟩ − \frac{2}{3}L_{aa}⟨f_{1;m}, f_{2}⟩ − \frac{2}{3}L_{aa}⟨f_{1}, f_{2;m}⟩ + ⟨(\frac{1}{12}L_{aa}L_{bb} − \frac{1}{6}L_{ab}L_{ab} +\frac{1}{6}R_{amam})f_{1}, f_{2}⟩\}dy.

(e) β_{4}(f_{1}, f_{2}, D, B^{−}) = \frac{1}{2}∫_{M}⟨Df_{1}, D̃f_{2}⟩ + ∫_{∂M}\{\frac{1}{2}⟨(Df_{1})_{;m}, f_{2}⟩ + \frac{1}{2}⟨f_{1}, (D̃f_{2})_{;m}⟩ −\frac{1}{4}⟨L_{aa}Df_{1}, f_{2}⟩ − \frac{1}{4}⟨L_{aa}f_{1}, D̃f_{2}⟩ + ⟨(\frac{1}{8}E_{;m} − \frac{1}{16}L_{ab}L_{ab}L_{cc} + \frac{1}{8}L_{ab}L_{ac}L_{bc} −\frac{1}{16}R_{ambm}L_{ab} + \frac{1}{16}R_{abcb}L_{ac} + \frac{1}{32}τ_{;m} + \frac{1}{16}L_{ab:ab})f_{1}, f_{2}⟩ − \frac{1}{4}L_{ab}⟨f_{1:a}, f_{2:b}⟩ −\frac{1}{8}⟨Ω_{am}f_{1:a}, f_{2}⟩ + \frac{1}{8}⟨Ω_{am}f_{1}, f_{2:a}⟩\}dy.

Theorem 2: (see [3, 7])

(a) β_{0}(f_{1}, f_{2}, D, B_{S}^{+}) = ∫_{M}⟨f_{1}, f_{2}⟩dx.

(b) β_{1}(f_{1}, f_{2}, D, B_{S}^{+}) = 0.

(c) β_{2}(f_{1}, f_{2}, D, B_{S}^{+}) = −∫_{M}⟨Df_{1}, f_{2}⟩dx + ∫_{∂M}⟨B_{S}^{+}f_{1}, f_{2}⟩dy.

(d) β_{3}(f_{1}, f_{2}, D, B_{S}^{+}) = \frac{2}{3} · 2π^{−1/2}∫_{∂M}⟨B_{S}^{+}f_{1}, B̃_{S}^{+}f_{2}⟩dy.

(e) β_{4}(f_{1}, f_{2}, D, B_{S}^{+}⟩ = \frac{1}{2}∫_{M}⟨Df_{1}, D̃f_{2}⟩dx + ∫_{∂M}\{−\frac{1}{2}⟨B_{S}^{+}f_{1}, D̃f_{2}⟩ −\frac{1}{2}⟨Df_{1}, B̃_{S}^{+}f_{2}⟩ + ⟨(\frac{1}{2}S + \frac{1}{4}L_{aa})B_{S}^{+}f_{1}, B̃_{S}^{+}f_{2}⟩dy.

(f) β_{5}(f_{1}, f_{2}, D, B_{S}^{+}) = 2π^{−1/2}∫_{∂M}\{−\frac{4}{15}⟨B_{S}^{+}Df_{1}, B̃_{S}^{+}f_{2}⟩ − \frac{4}{15}⟨B_{S}^{+}f_{1}, B̃_{S}^{+}D̃f_{2}⟩ −\frac{2}{15}⟨(B_{S}^{+}f_{1})_{:a}, (B̃_{S}^{+}f_{2})_{:a}⟩ + ⟨(\frac{2}{15}E + \frac{4}{15}S^{2} + \frac{4}{15}SL_{aa} + \frac{1}{30}L_{aa}L_{bb} + \frac{1}{15}L_{ab}L_{ab} −\frac{1}{15}R_{amam})B_{S}^{+}f_{1}, B̃_{S}^{+}f_{2}⟩\}dy.

(g) β_{6}(f_{1}, f_{2}, D, B_{S}^{+}) = −\frac{1}{6}∫_{M}⟨D^{2}f_{1}, D̃f_{2}⟩ + ∫_{∂M}\{\frac{1}{6}⟨B_{S}^{+}Df_{1}, D̃f_{2}⟩ +\frac{1}{6}⟨D^{2}f_{1}, B̃_{S}^{+}f_{2}⟩ + \frac{1}{6}⟨B_{S}^{+}f_{1}, D̃^{2}f_{2}⟩ − \frac{1}{6}⟨SB_{S}^{+}Df_{1}, B̃_{S}^{+}f_{2}⟩ − \frac{1}{6}⟨SB_{S}^{+}f_{1}, B̃_{S}^{+}D̃f_{2}⟩ −\frac{1}{12}⟨L_{aa}B_{S}^{+}Df_{1}, B̃_{S}^{+}f_{2}⟩ − \frac{1}{12}⟨L_{aa}B_{S}^{+}f_{1}, B̃_{S}^{+}D̃f_{2}⟩ + ⟨(\frac{1}{24}E_{;m} + \frac{1}{12}EL_{aa} +\frac{1}{48}L_{ab}L_{ab}L_{cc} + \frac{1}{24}L_{ab}L_{ac}L_{bc} − \frac{1}{48}R_{ambm}L_{ab} + \frac{1}{48}R_{abcb}L_{ac} − \frac{1}{24}R_{amam}L_{bb} +\frac{1}{96}τ_{;m} + \frac{1}{48}L_{ab:ab} + \frac{1}{12}SL_{aa}L_{bb} + \frac{1}{12}SL_{ab}L_{ab} − \frac{1}{12}SR_{amam} +\frac{1}{12}(SE + ES) + \frac{1}{4}S^{2}L_{aa} + \frac{1}{6}S^{3} + \frac{1}{6}S_{:aa})B_{S}^{+}f_{1}, B̃_{S}^{+}f_{2}⟩ −\frac{1}{12}L_{aa}⟨(B_{S}^{+}f_{1})_{:b}, (B̃_{S}^{+}f_{2})_{:b}⟩ − \frac{1}{12}L_{ab}⟨(B_{S}^{+}f_{1})_{:a}, (B̃_{S}^{+}f_{2})_{:b}⟩ −\frac{1}{6}⟨S(B_{S}^{+}f_{1})_{:a}, (B̃_{S}^{+}f_{2})_{:a}⟩ − \frac{1}{24}⟨Ω_{am}(B_{S}^{+}f_{1})_{:a}, B̃_{S}^{+}f_{2}⟩ +\frac{1}{24}⟨Ω_{am}B_{S}^{+}f_{1}, (B̃_{S}^{+}f_{2})_{:a}⟩\}dy. One can mix Neumann and Dirichlet boundary conditions. Near ∂M, let χ ∈

C^{∞}(End(V)) satisfy χ^{2} = 1 where χ is self-adjoint. Split V = V^{+} ⊕ V^{−} into the relevant eigenspaces; the trivial splitting V = V^{+} or V = V^{−} is permitted. Let Π^{±} be the associated projections on V^{±}. Let S ∈ C^{∞}(End(V^{+}|_{∂M})). Define mixed boundary conditions

B = B(χ, S, ∇^{D})f = {Π^{+}(∇_{m}^{D} + S)Π^{+}f}|_{∂M} ⊕ Π^{−}f|_{∂M}.

Absolute and relative boundary conditions for the DeRham complex are such boundary conditions; these are important in index theory. Let f^{±} = Π^{±}f.

Theorem 3: (see [7])

(a) β_{0}(f_{1}, f_{2}, D, B) = ∫_{M}⟨f_{1}, f_{2}⟩dx.

(b) β_{1}(f_{1}, f_{2}, D, B) = −2π^{−1/2}∫_{∂M}⟨f_{1}^{−}, f_{2}^{−}⟩dy.

(c) β_{2}(f_{1}, f_{2}, D, B) = −∫_{M}⟨Df_{1}, f_{2}⟩dx + ∫_{∂M}\{⟨f_{1;m}^{+} + Sf_{1}^{+}, f_{2}^{+}⟩ + ⟨\frac{1}{2}L_{aa}f_{1}^{−}, f_{2}^{−}⟩

−⟨f_{1}^{−}, f_{2;m}^{−}⟩}dy.

(d) β_{3}(f_{1}, f_{2}, D, B) = 2π^{−1/2}∫_{∂M}\{−\frac{2}{3}⟨f_{1;mm}^{−}, f_{2}^{−}⟩ − \frac{2}{3}⟨f_{1}^{−}, f_{2;mm}^{−}⟩

+\frac{2}{3}L_{aa}⟨f_{1;m}^{−}, f_{2}^{−}⟩ + \frac{2}{3}L_{aa}⟨f_{1}^{−}, f_{2;m}^{−}⟩ + ⟨(−\frac{1}{12}L_{aa}L_{bb} + \frac{1}{6}L_{ab}L_{ab}

−\frac{1}{6}R_{amam})f_{1}^{−}, f_{2}^{−}⟩ + \frac{2}{3}⟨f_{1;m}^{+} + Sf_{1}^{+}, f_{2;m}^{+} + Sf_{2}^{+}⟩ − ⟨Ef_{1}^{−}, f_{2}^{−}⟩

−\frac{2}{3}⟨Ef_{1}^{−}, f_{2}^{+}⟩ − \frac{2}{3}⟨Ef_{1}^{+}, f_{2}^{−}⟩ + ⟨f_{1}^{−}_{:a}, f_{2}^{−}_{:a}⟩ + \frac{2}{3}⟨f_{1}^{+}_{:a}, f_{2}^{−}_{:a}⟩

+\frac{2}{3}⟨f_{1}^{−}_{:a}, f_{2}^{+}_{:a}⟩\}dy.

The study of heat energy content asymptotics is relatively recent. If we set f_{1} = f_{2} = 1 and take D = Δ_{0} to be the scalar Laplacian, then

β(t) = β(1, 1, Δ_{0}, B^{−})(t) ≃ Σ_{n=0}^{∞}β_{n}(M)t^{n/2}

gives the asymptotic expansion of the total heat content of M with an initial temperature one. β_{0}(M) and β_{1}(M) for domains in Euclidean space were first computed by van den Berg and Davies [2]; β_{2}(M) was computed for domains in Euclidean space by van den Berg and Le Gall [4]; β_{0}(M), β_{1}(M), and β_{2}(M) were computed by van den Berg [1] for the upper hemisphere of a sphere; these results follow from Theorem 1. The case of polygonal domains in the plane was considered by van den Berg and Srisatkunarajah [6]; the analysis is quite different if the boundary is not smooth and these results do not follow from Theorem 1. We also refer to related work by Phillips and Jansons [9]. McAvity [8] first considered mixed boundary conditions and proved Theorem 3 using a modified DeWitt ansatz; we subsequently gave an independent derivation. Our results agree modulo notational changes. Acknowledgements: Research of P. Gilkey partially supported by the NSF and by Max Planck Institut für Mathematik

References

[1] M. van den Berg, Heat equation on a hemisphere, Proceedings of the Royal Soc. of Edinburgh 118A, 5-12 (1991).

[2] M. van den Berg and E. B. Davies, Heat flow out of regions in R^{m}, Math. Z. 202, 463-482 (1989).

[3] M. van den Berg, S. Desjardins, and P. Gilkey, Functorality and heat content asymptotics for operators of Laplace type (to appear).

[4] M. van den Berg and J-F Le Gall, Mean curvature and the heat equation (preprint).

[5] M. van den Berg and P. Gilkey, Heat content asymptotics of a Riemannian manifold with boundary (preprint).

[6] M. van den Berg and S. Srisatkunarajah, Heat flow and Brownian motion for a region in R^{2} with a polygonal boundary, Probab. Th. Rel. Fields 86, 41-52 (1990).

[7] S. Desjardins and P. Gilkey, Heat content asymptotics for operators of Laplace type with Neumann boundary conditions (preprint).

[8] D. M. McAvity, Heat kernel asymptotics for mixed boundary conditions, Class. Quant. Grav. 9 (1992), 1983., preprint DAMTP/92-17; see also Surface energy from heat content asymptotics to appear Journal of Physics A.

[9] C. G. Phillips and K. M. Jansons, The short-time transient of diffusion outside a conducting body, Proc. Royal Soc. London A 428, 431-449 (1990).

M. van den Berg, Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, U. K. e-mail: MTHMV@vaxa.heriot-watt.ac.uk S. Desjardins, Department of Mathematics, University of Oregon, Eugene Oregon 97403, USA. e-mail: desjard@bright.math.uoregon.edu. P. Gilkey, MPIM, 26 Gottfried Claren Str., 5300 Bonn 3 Germany; Department of Mathematics, University of Oregon, Eugene Oregon 97403, USA. e-mail: gilkey@bright.math.uoregon.edu.

Received 11 December 1992