Evolution of Ricci scalar under Finsler Ricci flow

TL;DR

This paper derives the evolution equations for Ricci scalar and curvature under Finsler Ricci flow, proving preservation of positivity and establishing bounds, thus advancing understanding of Finsler geometric flows.

Contribution

It introduces the evolution equations for Ricci scalar and reduced $hh$-curvature in Finsler Ricci flow and proves positivity preservation and bounds.

Findings

Ricci scalar evolution is a parabolic-type equation.

Positivity of flag curvature and Ricci scalar is preserved under flow.

A lower bound for Ricci scalar along the flow is established.

Abstract

Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, evolution equation of the reduced $hh$-curvature and the Ricci scalar along the Finslerian Ricci flow is obtained and it is proved that the Ricci flow preserves positivity of reduced $hh$-curvature on finite time. Next, it is shown that the evolution of Ricci scalar is a parabolic-type equation and if the initial Finsler metric is of positive flag curvature, then the flag curvature and the Ricci scalar remain positive as long as the solution exists. Finally, a lower bound for the Ricci scalar along the Ricci flow is obtained.