Exponential map and normal form for cornered asymptotically hyperbolic metrics

TL;DR

This paper develops a normal form and exponential map analysis for cornered asymptotically hyperbolic manifolds, extending geometric understanding near boundaries where finite and infinite parts intersect.

Contribution

It introduces a normal form at the corner for these manifolds, combining submanifold and asymptotic hyperbolic structures, and proves a Cartan-Hadamard type theorem for the exponential map.

Findings

Normal form construction at the corner

Cartan-Hadamard type theorem near infinity

Unified geometric analysis at boundary intersections

Abstract

This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary -- cornered asymptotically hyperbolic manifolds -- and proves a theorem of Cartan-Hadamard type near infinity for the normal exponential map on the finite boundary. As a main application, a normal form for such manifolds at the corner is then constructed, analogous to the normal form for usual asymptotically hyperbolic manifolds and suited to studying geometry at the corner. The normal form is at the same time a submanifold normal form near the finite boundary and an asymptotically hyperbolic normal form near the infinite boundary.