# The Riemannian Quantitative Isoperimetric Inequality

## TL;DR

This paper investigates the validity of the quantitative isoperimetric inequality on Riemannian manifolds, demonstrating its generic truth and establishing a sharp version for real analytic metrics without prior knowledge of minimizers.

## Contribution

It shows the Euclidean inequality fails generally on Riemannian manifolds but holds generically, and proves a sharp version for real analytic metrics using the Lojasiewicz-Simon inequality.

## Key findings

- The Euclidean quantitative isoperimetric inequality is false in general on Riemannian manifolds.
- The inequality holds generically on closed Riemannian manifolds.
- A sharp, modified inequality is established for real analytic metrics.

## Abstract

We study the Riemannian quantiative isoperimetric inequality. We show that direct analogue of the Euclidean quantitative isoperimetric inequality is--in general--false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) version of the quantitative isoperimetric inequality holds for a real analytic metric, using the Lojasiewicz-Simon inequality. A main novelty of our work is that in all our results we do not require any a priori knowledge on the structure/shape of the minimizers.