# Prescribing scalar curvatures: non compactness versus critical points at infinity

## TL;DR

This paper explores the behavior of a Yamabe-type flow on Riemannian manifolds, showing that certain prescribed scalar curvature functions lead to non-compact flow lines, while minor modifications can restore compactness.

## Contribution

It provides a specific example demonstrating the delicate balance between non-compactness and compactness in scalar curvature prescribing flows.

## Key findings

- Existence of non-compact flow lines for certain functions
- Minor modifications can induce flow compactness
- Insights into the geometric analysis of scalar curvature problems

## Abstract

We illustrate an example of a generic, positive function K on a Riemannian manifold to be conformally prescribed as the scalar curvature, for which the corresponding Yamabe type L2-gradient flow exhibits non compact flow lines, while a slight modification of it is compact.