Heat content asymptotics and immersions
by
Alessandro Savo
Universita' di Roma, La Sapienza, Italy

Oral Communication We fix a domain A in a Riemannian manifold M with metric g, and consider the following heat diffusion problem. Assume that A has constant unit temperature at time t=0, and that the boundary of A is kept at zero temperature at all times (this corresponds to imposing Dirichlet conditions on the boundary).

Then, heat will flow away from the domain and, as time goes to infinity, the temperature inside A will approach zero. The question is What is the heat content of the domain A at the time t? The heat content function of A, which we will denote by H(t), is given by the integral over A, for the Riemannian measure, of the corresponding temperature function, which is the solution of the heat equation (for the Laplacian of (M,g)) with Dirichlet boundary conditions and unit initial conditions.

One could ask what is the asymptotic behavior of the heat content for small or large times.

Due to the boundary refrigeration, the heat content will decrease exponentially fast to zero as time tends to infinity, and it is easy to see that it does so with a speed proportional to the first eigenvalue of the domain (with Dirichlet boundary conditions, of course).

For small times, the immersion of the boundary of A into our manifold comes into play, and it turns out that the heat content H(t) admits an asymptotic series in half-integral powers of t: Vol(A)+b(1)t^1/2+b(2)t+b(3)t^3/2+... As a consequence of P. Levy's "Principle of not feeling the boundary" one sees that the geometry of A, far from its boundary, will not affect the coefficient b(k); more precisely, if two domains are isometric near the respective boundaries, then all the heat content coefficients of the two domains will be the same.

Therefore each b(k) will be given by integration, over the boundary, of a certain invariant.

Up to a moltiplicative constant, b(1) is the volume of the boundary, and b(2) is the integral mean curvature of the boundary (for domains in R^n, this was first proved in [2]).

The coefficients b(3) and b(4) were later computed in [1]. The scope of this talk is to present an algorithm, developped in [3], which computes, by iteration, all the coefficients b(k). We do so by using the distance function from the boundary of the domain; then b(k) will be given by integration, over the boundary, and for the induced measure, of an explicit differential operator D(k) applied to the distance function. In turn, the operator D(k) is a homogeneous polynomial, of degree k, in only two operators: the Laplacian of (M,g), and the operator of order one given by the normal derivative minus multiplication by the mean curvature function. We then plan to examine the heat content coefficients for some special immersions, and possibly to see what happens when the boundary is polyhedral. REFERENCES [1] van den Berg, M., Gilkey, P.B. "Heat content asymptotics for a Riemannian manifold with boundary" J. Funct. Anal. 120 (1994), 48-71. [2] van den Berg, M., Le Gall, P.B. "Mean curvature and the heat equation" Math.Z. 215 (1994), 437-464. [3] Savo, A. " Uniform estimates and the whole asymptotic series of the heat content on manifolds"

Geom. Dedicata 73 (1998), 181-214.

Date received: June 15, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-26.