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October 2011 Ricci-flow-conjugated initial data sets for Einstein equations
Mauro Carfora
Adv. Theor. Math. Phys. 15(5): 1411-1484 (October 2011).

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.

Citation

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Mauro Carfora. "Ricci-flow-conjugated initial data sets for Einstein equations." Adv. Theor. Math. Phys. 15 (5) 1411 - 1484, October 2011.

Information

Published: October 2011
First available in Project Euclid: 10 October 2012

zbMATH: 1258.83008
MathSciNet: MR2989836

Rights: Copyright © 2011 International Press of Boston

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Vol.15 • No. 5 • October 2011
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