Geometric flows and applications

TL;DR

This paper explores how geometric flows, especially Ricci flows, relate to string theory and gravity, showing that certain gravitational instantons evolve similarly to geometric flows across dimensions.

Contribution

It demonstrates the correspondence between geometric flows and time evolution in string theory and gravity, extending to non-relativistic gravity under specific conditions.

Findings

Homogeneous, self-dual gravitational instantons evolve as geometric flows.

Time evolution in string theory can be approximated by Ricci-flow of spatial sections.

The property extends to non-relativistic gravity in any dimension under detailed-balance.

Abstract

I present some applications of geometric flows in string theory and gravity. In some circumstances time evolution in string theory can be approximately identified with Ricci-flow parametric evolution of spatial sections. In four dimensions, homogeneous, self-dual, gravitational instantons of general relativity evolve in time exactly as geometric flows of homogeneous three-manifolds. For non-relativistic versions of gravity, this property persists in any dimension, under the assumption of detailed-balance condition.