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

Type: Preprint

Publication Date: 2024-07-26

Citations: 0

DOI: https://doi.org/10.48550/arxiv.2407.18622

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 $K$.

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  • arXiv (Cornell University) - View - PDF

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