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

Abstract

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 .

Nonuniqueness for a fully nonlinear, degenerate elliptic boundary value problem in conformal geometry

“…In this paper we use bifurcation theory to give a nonuniqueness result for a fully nonlinear, degenerate elliptic boundary-value problem involving the σ k -curvature. Our result gives the first explicit examples of nonuniqueness for k = 4, and relies on the general bifurcation theorem proven by Case, Moreira and Wang [2]. We refer to the introduction of the article [2] for a thorough account of the history of this problem in the context of nonuniqueness results for Yamabe-type problems.…”

“…Our result gives the first explicit examples of nonuniqueness for k = 4, and relies on the general bifurcation theorem proven by Case, Moreira and Wang [2]. We refer to the introduction of the article [2] for a thorough account of the history of this problem in the context of nonuniqueness results for Yamabe-type problems.…”

“…One might wonder there is an infinite family of pairs (m, n) such that the Riemannian product S n × H m satisfies σ k = 0. This is shown in [2] to be the case if k ≤ 3. By running the Algcurves(genus) package in Maple, we see that the genus of {(m, n) : σ 4 (S n × H m ) = 0} is three.…”

“…S. Chen [3] introduced the invariant H 4 on the boundary so that the pair (σ 4 ; H 4 ) is variational. See [2] for more details.…”

“…The Case, Moreira and Wang [2] bifurcation theorem involves the following Dirichlet problem. Suppose T 3 := ∂σ4 ∂Ai,j is positive definite.…”