Ricci Flow Conjugated Initial Data Sets for Einstein Equations

Understand

We discuss a natural form of Ricci--flow conjugation between two distinct general relativistic data sets given on a compact $n\geq 3$-dimensional manifold $\Sigma$.

• We establish the existence of the relevant entropy functionals for the matter and geometrical variables, their monotonicity properties, and the associated convergence in the appropriate sense.
• We show that in such a framework there is a natural mode expansion generated by the spectral resolution of the Ricci conjugate Hodge--DeRham operator.
• This mode expansion allows to compare the two distinct data sets and gives rise to a computable heat kernel expansion of the fluctuations among the fields defining the data.