On compact Ricci solitons in Finsler geometry

TL;DR

This paper investigates the properties of Ricci solitons in Finsler geometry, establishing conditions for compactness, fundamental group finiteness, and cohomology vanishing, thus extending understanding of Ricci flow solutions in this setting.

Contribution

It proves that forward complete shrinking Ricci solitons are compact if bounded, and that compact shrinking Finsler Ricci solitons have finite fundamental groups and trivial first cohomology.

Findings

Shrinking Ricci solitons are compact iff bounded.

Compact shrinking Finsler Ricci solitons have finite fundamental group.

First de Rham cohomology vanishes for these solitons.

Abstract

Ricci solitons on Finsler spaces, previously developed by the present authors, are a generalization of Einstein spaces, which can be considered as a solution to the Ricci flow on compact Finsler manifolds. In the present work it is shown that on a Finslerian space, a forward complete shrinking Ricci soliton is compact if and only if it is bounded. Moreover, it is proved that a compact shrinking Finslerian Ricci soliton has finite fundamental group and hence the first de Rham cohomology group vanishes.