Lower bounds for warping functions on warped-product AHE manifolds

TL;DR

This paper establishes a lower bound for the warping function in warped-product asymptotically hyperbolic Einstein manifolds, linking geometric boundary conditions to interior metric properties.

Contribution

It provides the first lower bound for the warping function based solely on boundary conformal data in warped-product AHE manifolds.

Findings

Positive Yamabe constant on boundary implies positive lower bound for warping function

Lower bound depends only on boundary conformal class

Results connect boundary geometry to interior metric properties

Abstract

Let $[\gamma]$ be the conformal boundary of a warped product $C^{3,\alpha}$ AHE metric $g=g_M+u^2h$ on $N=M \times F$, where $(F,h)$ is compact with unit volume and nonpositive curvature. We show that if $[\gamma]$ has positive Yamabe constant, then $u$ has a positive lower bound that depends only on $[\gamma]$.