# Integral of scalar curvature on non-parabolic manifolds

## TL;DR

This paper establishes a bound on the asymptotic weighted integral of scalar curvature for complete, non-parabolic 3-manifolds with non-negative Ricci curvature, linking it to the asymptotic volume ratio.

## Contribution

It introduces a new bound on the scalar curvature integral for non-parabolic manifolds using monotonicity formulas, connecting geometric analysis with volume growth.

## Key findings

- Bound on scalar curvature integral in terms of volume ratio
- Application of monotonicity formulas of Colding and Minicozzi
- Results specific to 3-dimensional non-parabolic manifolds

## Abstract

Using the monotonicity formulas of Colding and Minicozzi, we prove that on any complete, non-parabolic Riemannian manifold $(M^3, g)$ with non-negative Ricci curvature, the asymptotic weighted scaling invariant integral of scalar curvature has an explicit bound in form of asymptotic volume ratio.