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We find exact solutions describing Ricci flows of four-dimensional pp-waves nonlinearly deformed by two-/three-dimensional solitons. Such solutions are parametrized by five-dimensional metrics with generic off-diagonal terms and connections with nontrivial torsion which can be related, for instance, to antisymmetric tensor sources in string gravity. There are defined nontrivial limits to four-dimensional configurations and the Einstein gravity.

In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four-dimensional (4D) Taub-NUT space–times. It is outlined a new geometric technique of constructing Ricci flow solutions. Some conceptual issues on space–times provided with generic off-diagonal metrics and associated nonlinear connection structures are analyzed. The limit from gravity/Ricci flow models with nontrivial torsion to configurations with the Levi-Civita connection is allowed in some specific physical circumstances by constraining the class of integral varieties for the Einstein and Ricci flow equations.

The idea is considered that a quantum wormhole in a spacetime foam can be described as a Ricci flow. In this interpretation, the Ricci flow is a statistical system and every metric in the Ricci flow is a microscopical state. The probability density of the microscopical state is connected with a Perelman's functional of a rescaled Ricci flow.

Stochastic Einstein equations are considered when three-dimensional space metric γ_ij are stochastic functions. The probability density for the stochastic quantities is connected with Perelman's entropy functional. As an example, the Friedman Universe is considered. It is shown that for the Friedman Universe the dynamical evolution is not changed. The connection between general relativity and Ricci flow is discussed.

There were elaborated different models of Finsler geometry using the Cartan (metric compatible), or Berwald and Chern (metric non-compatible) connections, the Ricci flag curvature, etc. In a series of works, we studied (non)-commutative metric compatible Finsler and non-holonomic generalizations of the Ricci flow theory [see S. Vacaru, J. Math. Phys. 49 (2008) 043504; 50 (2009) 073503 and references therein]. The aim of this work is to prove that there are some models of Finsler gravity and geometric evolution theories with generalized Perelman's functionals, and correspondingly derived non-holonomic Hamilton evolution equations, when metric non-compatible Finsler connections are involved. Following such an approach, we have to consider distortion tensors, uniquely defined by the Finsler metric, from the Cartan and/or the canonical metric compatible connections. We conclude that, in general, it is not possible to elaborate self-consistent models of geometric evolution with arbitrary Finsler metric non-compatible connections.

It is proved that given a conformal metric e^u0g₀, with e^u0 ∈ L^∞, on a 2-dim closed Riemannian manfold (M, g₀), there exists a unique smooth solution u(t) of the Ricci flow such that u(t) → u₀ in L² as t → 0.