On the Nirenberg problem on spheres: Arbitrarily many solutions in a perturbative setting

TL;DR

This paper proves that on spheres, for generic functions close to the scalar curvature of the standard metric, there are arbitrarily many conformal metrics with prescribed scalar curvature, using Morse theory and blow-up analysis.

Contribution

It introduces a new Morse-theoretic approach to establish multiple solutions for the prescribed scalar curvature problem without symmetry assumptions.

Findings

Existence of arbitrarily many solutions under generic conditions.

Characterization of blow-up solutions for a subcritical approximation.

Application of Morse relations to count solutions.

Abstract

Given a smooth positive function $K$ on the standard sphere $(\mathbb{S}^n,g_0)$, we use Morse theoretical methods and counting index formulae to prove that, under generic conditions on the function $K$, there are arbitrarily many metrics $g$ conformally equivalent to $g_0$ and whose scalar curvature is given by the function $K$ provided that the function is sufficiently close to the scalar curvature of $g_0$. Our approach leverages a comprehensive characterization of blowing-up solutions of a subcritical approximation, along with various Morse relations involving their indices. Notably, this multiplicity result is achieved without relying on any symmetry or periodicity assumptions about the function $K$.