Ricci-flow-conjugated initial data sets for Einstein equations. (English) Zbl 1258.83008

Carfora, Mauro

Adv. Theor. Math. Phys. 15, No. 5, 1411-1484 (2011).

Summary: 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 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 Ricci-flow 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.

MSC:

83C05	Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53Z05	Applications of differential geometry to physics
35L15	Initial value problems for second-order hyperbolic equations
94A17	Measures of information, entropy

Keywords:

Ricci-flow; Hodge-DeRham operator; entropic stability