Ricci-flow-conjugated initial data sets for Einstein equations

Abstract

We discuss a natural form of Ricci-flow conjugation between two distinct general relativistic data sets given on a compact $n \ge 3$-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 one 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. In particular, this shows that Ricciflow conjugation entails a natural form of $L^2$ parabolic averaging of one data set with respect to the other with a number of desirable properties: (i) It preserves the dominant energy condition; (ii) It is localized by a heat kernel whose support sets the scale of averaging; (iii) It is characterized by a set of balance functionals, that allow the analysis of its entropic stability.