Bifurcation of periodic solutions to the singular Yamabe problem on spheres

Abstract

We obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere, that blow up along a great circle. These are (complete) constant scalar curvature metrics on the complement of S 1 inside S m , m ≥ 5, that are conformal to the round (incomplete) metric and periodic in the sense of being invariant under a discrete group of conformal transformations. These solutions come from bifurcating branches of constant scalar curvature metrics on compact quotients of S m \ S 1 ∼ = S m−2 × H 2 .

Multiplicity of constant scalar curvature metrics in Tk×M

“…Motivation for the above result was given by the recent paper [9], where a similar bifurcation problem is studied in the context of singular Yamabe solutions in spheres that blowup on an equator. In that case, the authors consider products of the type Σ × M, where Σ is an orientable closed surface of genus greater than 1, endowed with a metric varying in the set of hyperbolic metrics (i.e., with curvature equal to −1).…”

“…The coefficients of ∆ r are expressed in terms of the metric coefficients g ij t r , see (9), and they tend uniformly to the coefficients of the elliptic operator ∆ t * . Hence, in (13) we can choose coefficients C r ≡ C and σ r ≡ σ that do not depend on r, and obtain that u r tends to 1 in W 2,p , which proves our claim.…”

Infinitely many solutions to the Yamabe problem on noncompact manifolds

“…Given a closed manifold (M, g) and a closed subset Λ ⊂ M , the singular Yamabe problem consists of finding a complete metric g ′ on M \ Λ that has constant scalar curvature and is conformal to g. In other words, these are solutions to the Yamabe problem on M that blow up on Λ. Consider the case in which (M, g) is the round sphere (S m , g round ) and Λ = S k is a round subsphere, which was also studied in [10,31,32,38]. There is a conformal equivalence…”

“…given by first using the stereographic projection with a point in S k to obtain a conformal equivalence with (R m \R k , g flat ), and second using cylindrical coordinates g flat = dr 2 + r 2 dθ 2 + dy 2 to conclude that 1 r 2 g flat = g round ⊕ g hyp , see also [10,32]. The conformal equivalence (3.2) provides a trivial solution f * (g round ⊕ g hyp ) to the singular Yamabe problem on S m \ S k , with constant scalar curvature equal to scal m,k = (m − 2k − 2)(m − 1).…”

“…If k > (m − 2)/2, then scal m,k < 0 and this is the unique solution by an argument involving the asymptotic maximum principle [31]. Existence of infinitely many (nonisometric) periodic solutions if Λ = S 1 is a great circle, i.e., k = 1, m ≥ 5, was recently obtained using bifurcation techniques [10].…”