Mass functions of a compact manifold

TL;DR

This paper introduces mass functions for compact manifolds, linking the supremum and infimum of metric masses with Yamabe constants, and explores their properties and applications to the Yamabe invariant.

Contribution

It defines and analyzes the properties of mass functions related to Yamabe constants on compact manifolds, providing new tools for geometric analysis.

Findings

Mass functions are well-defined and possess key properties.

These functions relate to the Yamabe invariant of the manifold.

Applications include bounds and characterizations of the Yamabe invariant.

Abstract

Let $M$ be a compact manifold of dimension $n$. In this paper, we introduce the {\em Mass Function} $a \geq 0 \mapsto \xp{M}{a}$ (resp. $a \geq 0 \mapsto \xm{M}{a}$) which is defined as the supremum (resp. infimum) of the masses of all metrics on $M$ whose Yamabe constant is larger than $a$ and which are flat on a ball of radius~$1$ and centered at a point $p \in M$. Here, the mass of a metric flat around~$p$ is the constant term in the expansion of the Green function of the conformal Laplacian at~$p$. We show that these functions are well defined and have many properties which allow to obtain applications to the Yamabe invariant (i.e. the supremum of Yamabe constants over the set of all metrics on $M$).