Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres

DOI10.1007/s00526-012-0535-yzbMath1272.53042arXiv1107.5335OpenAlexW3098987586MaRDI QIDQ353116

Paolo Piccione, Renato G. Bettiol

Publication date: 12 July 2013

Published in: Calculus of Variations and Partial Differential Equations (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1107.5335

zbMATH Keywords

Riemannian submersionYamabe problemEinstein metricconformal classBerger sphereHilbert-Einstein functionalhomogeneous sphere

Mathematics Subject Classification ID

Differential geometry of homogeneous manifolds (53C30) Global Riemannian geometry, including pinching (53C20) Rigidity results (53C24) Applications of variational problems in infinite-dimensional spaces to the sciences (58E50) Variational problems in abstract bifurcation theory in infinite-dimensional spaces (58E07) Bifurcation theory for PDEs on manifolds (58J55)

Related Items

Metrics of constant scalar curvature on sphere bundles ⋮ Rigidity and bifurcation results for CMC hypersurfaces in warped product spaces ⋮ Infinitely many solutions to the Yamabe problem on noncompact manifolds ⋮ Bifurcation of metrics with null scalar curvature and constant mean curvature on the boundary of compact manifolds ⋮ Bifurcation for the constant scalar curvature equation and harmonic Riemannian submersions ⋮ Multipeak solutions for the Yamabe equation ⋮ Multiplicity of constant scalar curvature metrics in \(\mathbb{T}^k \times M\) ⋮ Full Laplace spectrum of distance spheres in symmetric spaces of rank one ⋮ Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory ⋮ Diameter and Laplace eigenvalue estimates for left-invariant metrics on compact Lie groups ⋮ Harmonically free group actions and equivariant bifurcation in the Yamabe problem ⋮ Global bifurcation techniques for Yamabe type equations on Riemannian manifolds ⋮ Degenerate solutions of a nonlinear elliptic equation on the sphere ⋮ Multiplicity results for the Yamabe equation by Lusternik-Schnirelmann theory ⋮ On the bifurcation of solutions of the Yamabe problem in product manifolds with minimal boundary ⋮ Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on maximal flag manifolds ⋮ Uniqueness of solutions of the Yamabe problem on manifolds with boundary ⋮ The first eigenvalue of a homogeneous CROSS ⋮ Equivariant Bifurcation in Geometric Variational Problems ⋮ On uniqueness and existence of conformally compact Einstein metrics with homogeneous conformal infinity

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• On bifurcation of solutions of the Yamabe problem in product manifolds
• On the spectrum of Riemannian submersions with totally geodesic fibers
• Analytic vectors
• Compactness of solutions to the Yamabe problem. III
• A compactness theorem for the Yamabe problem
• Blow-up phenomena for the Yamabe equation. II
• Conformal deformation of a Riemannian metric to constant scalar curvature
• Some metrics on a (4r+3)-sphere and spectra
• Homogeneous Einstein metrics on spheres and projective spaces
• Laplacians and Riemannian submersions with totally geodesic fibres
• Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire
• The first eigenvalue of the Laplacian on spheres
• Bifurcation theory. An introduction with applications of PDEs
• Nonuniqueness and high energy solutions for a conformally invariant scalar equation
• A multiplicity result for the Yamabe problem on \(S^n\)
• A priori estimates for the Yamabe problem in the non-locally conformally flat case
• Le spectre d'une variété riemannienne. (The spectrum of a Riemannian manifold)
• The conjectures on conformal transformations of Riemannian manifolds
• Compactness of solutions to the Yamabe problem. II
• Bifurcation and symmetry-breaking
• A note on the uniqueness of solutions for the Yamabe problem
• Strongly Commuting Selfadjoint Operators and Commutants of Unbounded Operator Algebras
• ON UNIQUENESS AND DIFFERENTIABILITY IN THE SPACE OF YAMABE METRICS
• Non-compactness and multiplicity results for the Yamabe problem on \(S^n\)
• Convergence of Alexandrov spaces and spectrum of Laplacian