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

2024/07/26 by Ahmedou, Mohameden, Ayed, Mohamed Ben, Mehdi, Khalil El
#35A01 #58E05 #58J05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.18622

Abstract

Given a smooth positive function K on the standard sphere (\mathbbSⁿ,g₀), 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 g₀ and whose scalar curvature is given by the function K provided that the function is sufficiently close to the scalar curvature of g₀. 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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