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This book contains a self-consistent treatment of a geometric
averaging technique, induced by the Ricci flow, that allows
comparing a given (generalized) Einstein initial data set with
another distinct Einstein initial data set, both supported on a
given closed n-dimensional manifold. This is a case study where two
vibrant areas of research in geometric analysis, Ricci flow and
Einstein constraints theory, interact in a quite remarkable way.
The interaction is of great relevance for applications in
relativistic cosmology, allowing a mathematically rigorous approach
to the initial data set averaging problem, at least when data sets
are given on a closed space-like hypersurface. The book does not
assume an a priori knowledge of Ricci flow theory, and considerable
space is left for introducing the necessary techniques. These
introductory parts gently evolve to a detailed discussion of the
more advanced results concerning a Fourier-mode expansion and a
sophisticated heat kernel representation of the Ricci flow, both of
which are of independent interest in Ricci flow theory. This work
is intended for advanced students in mathematical physics and
researchers alike.
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