Convergence of Finslerian metrics under Ricci flow

TL;DR

This paper proves that solutions to the Finslerian Ricci flow on compact manifolds converge smoothly to a limit metric as time approaches a finite singularity, and shows curvature blow-up in short time.

Contribution

It establishes convergence of Finslerian metrics under Ricci flow and analyzes curvature behavior near singularities.

Findings

Finslerian Ricci flow solutions converge in $C^{ abla}$ to a smooth limit

Curvature tensor blows up in short time on compact manifolds

Finite-time convergence of Finslerian metrics under Ricci flow

Abstract

In this work, convergence of evolving Finslerian metrics first in a general flow next under Finslerian Ricci flow is studied. More intuitively it is proved that a family of Finslerian metrics $g(t)$ which are solutions to the Finslerian Ricci flow converge in $C^{\infty}$ to a smooth limit Finslerian metric as $ t $ approaches the finite time $ T $. As a consequence of this result one can show that in a compact Finsler manifold the curvature tensor along Ricci flow blows up in short time.