Symmetry Reduction of the Two-Dimensional Ricci Flow Equation

TL;DR

This paper classifies symmetries of the 2D Ricci flow equation, reduces it using similarity methods, and finds new invariant solutions, advancing understanding of its geometric evolution.

Contribution

It provides a complete symmetry classification and derives new invariant solutions for the 2D Ricci flow equation using Lie group analysis.

Findings

Classification of the symmetry group of the 2D Ricci flow equation

Derivation of reduced equations via similarity reduction

New explicit invariant solutions obtained for the equation

Abstract

This paper is devoted to obtain the one-dimensional group invariant solutions of the two-dimensional Ricci flow ((2D) Rf) equation. By classifying the orbits of the adjoint representation of the symmetry group on its Lie algebra, the optimal system of one-dimensional subalgebras of the ((2D) Rf) equation is obtained. For each class, we will find the reduced equation by the method of similarity reduction. By solving these reduced equations, we will obtain new sets of group invariant solutions for the ((2D) Rf) equation.