  This paper is divided into three parts. The first part focuses on periodic
layer heat potentials, demonstrating their smooth dependence on regular
perturbations of the support of integration. In the second part, we present an
application of the results from the first part. Specifically, we consider a
transmission problem for the heat equation in a periodic two-phase composite
material and we show that the solution depends smoothly on the shape of the
transmission interface, boundary data, and conductivity parameters. Finally, in
the last part of the paper, we fix all parameters except for the contrast
parameter and outline a strategy to deduce an explicit expansion of the
solution using a Neumann-type series.
