Preprint A444/2006
On Yamabe constants of Riemannian products
Jimmy Petean | Akutagawa, Kazuo | Florit, Luis A.
Keywords: Yamabe constant | product manifold
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 x N, g + r h) as r goes to infinity is equal to the Yamabe constant of (M^m x R^n, [g + g_0]) and is strictly less than the Yamabe invariant of the sphere S^{m+n} for n > 1. 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.