Solutions of the Yamabe Equation by Lyapunov–Schmidt Reduction

• Published in: The Journal of geometric analysis 2021-08, Vol.31 (8), p.8080-8104
• Main Authors: Dávila, Jorge, Munive, Isidro H.
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Mathematics; Mathematics and Statistics; Reduction; Riemann manifold; Riemann surfaces
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-020-00570-4

Given any closed Riemannian manifold ( M ,  g ) we use the Lyapunov–Schmidt finite-dimensional reduction method and the classical Morse and Lusternick–Schnirelmann theories to prove multiplicity results for positive solutions of a subcritical Yamabe type equation on ( M ,  g ). If ( N ,  h ) is a closed Riemannian manifold of constant positive scalar curvature we obtain multiplicity results for the Yamabe equation on the Riemannian product ( M × N , g + ε 2 h ) , for ε > 0 small. For example, if M is a closed Riemann surface of genus g and ( N , h ) = ( S 2 , g 0 ) is the round 2-sphere, we prove that for ε > 0 small enough and a generic metric g on M , the Yamabe equation on ( M × S 2 , g + ε 2 g 0 ) has at least 2 + 2 g solutions.