Some formulas of Santal\'o type in Finsler geometry and its applications

TL;DR

This paper develops Santaló type formulas in Finsler geometry and applies them to derive eigenvalue bounds, isoperimetric inequalities, and finiteness theorems, advancing the understanding of geometric analysis in Finsler spaces.

Contribution

It introduces two new Santaló type formulas for general Finsler manifolds, enabling several significant geometric and spectral results.

Findings

Universal lower bound for the first eigenvalue of the nonlinear Laplacian

Two Croke type isoperimetric inequalities

Yamaguchi type finiteness theorem in Finsler geometry

Abstract

In this paper, we establish two Santal\'o type formulas for general Finsler manifolds. As applications, we derive a universal lower bound for the first eigenvalue of the nonlinear Laplacian, two Croke type isoperimetric inequalities, and a Yamaguch type finiteness theorem in Finser geometry.