# Off-diagonal cosmological solutions in emergent gravity theories and Grigory Perelman entropy for geometric flows

## TL;DR

This paper develops a framework linking relativistic geometric flows, emergent gravity, and entropy functionals, leading to new exact solutions for cosmological models with complex structures and potential insights into structure formation.

## Contribution

It introduces a novel approach connecting Perelman's entropy functionals with relativistic geometric flows and emergent gravity, deriving new exact solutions in modified gravity and cosmology.

## Key findings

- Derived relativistic geometric evolution equations using entropy functionals.
- Constructed new classes of exact solutions for anisotropic cosmological metrics.
- Proposed models explaining structure formation via off-diagonal solutions in entropic gravity.

## Abstract

We develop an approach to the theory of relativistic geometric flows and emergent gravity defined by entropy functionals and related statistical thermodynamics models. Nonholonomic deformations of G. Perelman's functionals and related entropic values are used for deriving relativistic geometric evolution flow equations. For self-similar configurations, such equations describe generalized Ricci solitons defining modified Einstein equations. We analyze possible connections between relativistic models of nonholonomic Ricci flows and emergent modified gravity theories. We prove that corresponding systems of nonlinear partial differential equations, PDEs, for entropic flows and modified gravity possess certain general decoupling and integration properties. There are constructed new classes of exact and parametric solutions for nonstationary configurations and locally anisotropic cosmological metrics in modified gravity theories and general relativity. Such solutions describe scenarios of nonlinear geometric evolution and gravitational and matter field dynamics with pattern-forming and quasiperiodic structure and various space quasicrystal and deformed spacetime crystal models. We analyze new classes of generic off-diagonal solutions for entropic gravity theories and show how such solutions can be used for explaining structure formation in modern cosmology. Finally, we speculate why the approaches with Perelman-Lyapunov type functionals are more general or complementary to the constructions elaborated using the concept of Bekenstein-Hawking entropy.