Harmonically free group actions and equivariant bifurcation in the Yamabe problem

Moreira, Ana Claudia

doi:10.1016/j.difgeo.2017.09.001

Cited by 2 publications

(1 citation statement)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It was studied by Escobar [20], and recently by Brendle and Chen [9]. Moreover, nonuniqueness when the interior dimension is at least three has recently been established using bifurcation theory both in the minimal boundary [17,33] and the scalar flat [16] cases. In the latter case, a direct study of the underlying PDE has been carried out by de la Torre, del Pino, González and Wei [15].…”

Section: Introductionmentioning

confidence: 99%

Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

2019

Self Cite

View full text Add to dashboard Cite

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing σ k -curvature in the interior and constant H k -curvature on the boundary. When restricting to the closure of the positive k-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem which allows us to construct examples of compact Riemannian manifolds (X, g) for which this problem admits multiple nonhomothetic solutions in the case when 2k < dim X. Our examples are all such that the boundary with its induced metric is a Riemannian product of a round sphere with an Einstein manifold.2010 Mathematics Subject Classification. Primary 58J32; Secondary 53A30, 58J40.Thus depending on the different volume constraints, there are two types of boundary Yamabe problem. If one considers,n−1 n , then the Euler-Lagrangian equation (with suitable normalization) is given byThis boundary Yamabe problem is called the scalar flat type. This problem, as remarked by Escobar in [19], is a higher dimensional generalization of the Riemann mapping theorem. It was studied by Escobar [19,21] and Marques [31,32]. If one considers instead inf g∈[g0] X J g dvol g + M H g dvol ι * g Vol g (X) n−1 n+1, then the Euler-Lagrangian equation is given by R g = const, in X, H g = 0, on M .

show abstract

Section: Introductionmentioning

confidence: 99%

Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

2019

Self Cite

View full text Add to dashboard Cite

show abstract

Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

Case

Moreira

Wang

2018

Preprint

View full text Add to dashboard Cite

Harmonically free group actions and equivariant bifurcation in the Yamabe problem

Abstract: We study multiplicity of constant scalar curvature metrics in products of a compact closed manifold and a compact manifold with boundary using equivariant bifurcation theory.

Cited by 2 publications

References 22 publications

Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

Contact Info

Product

Resources

About