On Yamabe constants of Riemannian products

Abstract

Abstract. For a closed Riemannian manifold (M m , g) of constant positive scalar curvature and any other closed Riemannian manifold (N n , h), we show that the limit of the Yamabe constants of the Riemannian products (M × N, g + rh) as r goes to infinity is equal to the Yamabe constant of (M m × R n , [g + g E ]) and is strictly less than the Yamabe invariant of S m+n provided n ≥ 2. We then consider the minimum of the Yamabe functional restricted to functions of the second variable and we compute the limit in terms of the best constants of the Gagliardo-Nirenberg inequalities.

Isoperimetric regions in spherical cones and Yamabe constants of M × S 1

“…The fact that Y (S n × S 1 ) = Y n+1 is one of the first things we learned about the Yamabe invariant [11,22]. One way to see this is as follows: first one notes that lim T →∞ Y (S n × S 1 , [1] (the Yamabe constant for a non-compact Riemannian manifold is computed as the infimum of the Yamabe functional over compactly supported functions). But the Yamabe function for g 0 + dt 2 is precisely the conformal factor between S n × R and S n+1 − {S, N }.…”

“…It is a corollary to the previous theorem that this is actually true in the case n = 1. Namely, using the notation (as in [1])…”

Aubin’s Lemma for the Yamabe constants of infinite coverings and a positive mass theorem

“…[1][2][3][4]6,7,39]). In this article, we will present some extensions to Aubin's Lemma which will include, as a particular case, the first equality in the above working hypothesis (4). Therefore, by combining this with the inequality (3), the second inequality in (4) follows.…”

“…Then, using a similar argument to that in the proof of [2, Theorem 6.13] combined with the properties for G ∞ , G k in Lemma 2.1, we obtain (cf. [4])…”

A note on solutions of Yamabe-type equations on products of spheres

“…Interest in finding all solutions of the Yamabe equation in this case comes from trying to compute the Yamabe constants and its limit lim δ→0 Y (S n ×S n , [g n 0 +δg n 0 ]) = Y (S n ×R n , g n 0 +dx 2 ) (see [2]). An important question raised in [2,3], related to the computations of these Yamabe constants, is whether all solutions of the Yamabe equation on certain Riemannian products, like products of spheres (or the product of a sphere with Euclidean space), depend on only one of the factors. The main goal of this article is to show the existence of solutions which depend non-trivially on both factors: positive solutions when p < p n and nodal solutions when p = p n .…”