Advances in Theoretical and Mathematical Physics
Volume 15 (2011)
Ricci-flow-conjugated initial data sets for Einstein equations
Pages: 1411 – 1484
We discuss a natural form of Ricci-flow conjugationbetween two distinct general relativistic data sets givenon a compact $n\geq 3$-dimensional manifold $\Sigma$. Weestablish the existence of the relevant entropy functionalsfor the matter and geometrical variables, theirmonotonicity properties, and the associated convergence inthe appropriate sense. We show that in such a frameworkthere is a natural mode expansion generated by the spectralresolution of the Ricci conjugate Hodge--DeRham operator.This mode expansion allows one to compare the two distinct datasets and gives rise to a computable heat kernel expansionof the fluctuations among the fields defining the data. Inparticular, this shows that Ricci-flow conjugation entails anatural form of $L^2$ parabolic averaging of one data setwith respect to the other with a number of desirableproperties: (i) It preserves the dominant energycondition; (ii) It is localized by a heat kernelwhose support sets the scale of averaging; (iii) Itis characterized by a set of balance functionals, thatallow the analysis of its entropic stability.