Bounds of Scalar curvature, S-curvature and distortion on $\infty$-Einstein Finsler manifolds

TL;DR

This paper explores curvature bounds and topological properties of $ abla$-Einstein Finsler manifolds, establishing inequalities and finiteness results that advance understanding of Finsler geometric analysis and relate to Gromov's scalar curvature conjecture.

Contribution

It introduces new bounds for S-curvature, distortion, and scalar curvature on $ abla$-Einstein Finsler manifolds, linking curvature conditions to topological finiteness and addressing Gromov's conjecture.

Findings

Established linear growth lower bounds for S-curvature and distortion.

Derived bounds for scalar curvature on gradient Ricci solitons.

Proved topological finiteness for certain $ abla$-Einstein Finsler manifolds.

Abstract

This manuscript investigates the curvature and topological properties of certain $\infty$-Einstein Finsler metrics on Finsler metric measure spaces. By imposing symmetry conditions, we construct a series of special metrics and analyze their equivalence on special manifolds. Provided a Ricci curvature bound, we establish a linear growth lower bound estimate for the S-curvature and the distortion, revealing the interplay between curvature and measure on $\infty$-Einstein Finsler manifolds. Furthermore, by introducing scalar curvature and imposing a linear growth lower bound condition, we derive upper and lower bounds for the distortion, S-curvature, and the scalar curvature itself on asymmetric essential gradient Ricci solitons with certain non-Riemannian curvature constraints. These results yield direct topological finiteness conclusions for some forward-complete $\infty$-Einstein Finsler manifolds. Our work partially addresses Gromov's conjecture of scalar curvature in the context of Finsler metric measure spaces and provides a foundation for further research in geometric analysis within general Finsler geometry.