Volume growth on manifolds with more than one end

TL;DR

This paper extends the construction of metrics with prescribed volume growth to manifolds with multiple ends, providing bounds and analyzing properties like volume doubling and R.C.A. for geometric analysis applications.

Contribution

It generalizes previous results to manifolds with multiple ends and introduces Grimaldi-Pansu metrics with uniform volume growth bounds.

Findings

Constructed Grimaldi-Pansu metrics on multi-ended manifolds.

Established uniform bounds for volume growth functions.

Analyzed volume doubling and R.C.A. properties of these metrics.

Abstract

For an open manifold $M$ and a function $v$ with bounded growth of derivative, there exists a Riemannian metric of bounded geometry on $M$ such that the volume growth function lies in the same growth class as $v$. This was proved by R. Grimaldi and P. Pansu with the proof focusing on the case of manifolds with a single end. We prove this in the case of manifolds with multiple ends and call the constructed metrics Grimaldi-Pansu metrics. We give uniform bounds for the volume growth function of these metrics in terms of the given bgd-function in the case of a certain class of manifolds which can be written as connected sums of a finite collection of closed and compact manifolds. We study the volume doubling condition and the Relatively Connected Annulus (R.C.A.) property of the Grimaldi-Pansu metrics, which play an important role in studying geometric analysis on manifolds with finitely many ends.