Second Yamabe constant on Riemannian products

Abstract

Let ( M m , g ) be a closed Riemannian manifold ( m ≥ 2 ) of positive scalar curvature and ( N n , h ) any closed manifold. We study the asymptotic behaviour of the second Yamabe constant and the second N -Yamabe constant of ( M × N , g + t h ) as t goes to + ∞ . We obtain that lim t → + ∞ Y 2 ( M × N , [ g + t h ] ) = 2 2 m + n Y ( M × R n , [ g + g e ] ) . If n ≥ 2 , we show the existence of nodal solutions of the Yamabe equation on ( M × N , g + t h ) (provided t large enough). When s g is constant, we prove that lim t → + ∞ Y N 2 ( M × N , g + t h ) = 2 2 m + n Y R n ( M × R n , g + g e ) . Also we study the second Yamabe invariant and the second N -Yamabe invariant.