Two theorems on the intersections of horospheres in asymptotically harmonic spaces

TL;DR

This paper investigates the geometric properties of horospheres in asymptotically harmonic spaces, establishing theorems on their intersections, volume bounds, and mappings that preserve certain distances under specific conditions.

Contribution

It introduces new theorems on the intersection behavior of horospheres and constructs volume-preserving mappings in asymptotically harmonic manifolds.

Findings

Volume of horosphere intersections is independent of Busemann function differences.

Constructed mappings preserve distances in some directions under visibility condition.

Proved integrals over horosphere intersections are invariant to Busemann function differences.

Abstract

We use Busemann functions to construct volume preserving mappings in an asymptotically harmonic manifold. If the asymptotically harmonic manifold satisfies the visibility condition, we construct mappings which preserve distances in some directions. We also prove that some integrals on the intersection of horospheres are independent of the differences between the values of the corresponding Busemann functions and we establish an upper bound of the volume of the intersection of two horospheres which is independent of the difference between values of corresponding Busemann functions.