# Prescribing Morse scalar curvatures: critical points at infinity

## TL;DR

This paper investigates the problem of prescribing scalar curvature on closed Riemannian manifolds, establishing the equivalence of subcritical approximation and pseudo-gradient flow methods under certain conditions, and characterizing solutions at infinity.

## Contribution

It demonstrates the equivalence of two main analytical approaches for solving the scalar curvature prescription problem and characterizes the solutions at infinity under a mild non-degeneracy assumption.

## Key findings

- Equivalence of subcritical approximation and pseudo-gradient flow methods.
- Characterization of solutions at infinity and their relation to critical points.
- Identification of conditions under which solutions correspond to critical points with negative Laplacian.

## Abstract

The problem of prescribing conformally the scalar curvature of a closed Riemannian manifold as a given Morse function reduces to solving an elliptic partial differential equation with critical Sobolev exponent. Two ways of attacking this problem consist in subcritical approximations or negative pseudo gradient flows. We show under a mild none degeneracy assumption the equivalence of both approaches with respect to zero weak limits, in particular an one to one correspondence of zero weak limit finite energy subcritical blow-up solutions, zero weak limit critical points at infinity of negative type and sets of critical points with negative Laplacian of the function to be prescribed.