Second Yamabe constant on Riemannian products.

GUILLERMO HENRY

JOURNAL OF GEOMETRY AND PHYSICS

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2017 vol. 114 p. 260 - 275

0393-0440

Let $(M^m,g)$ be a closed Riemannian manifold  $(mgeq 2)$ 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 $(Mimes N,g+th)$ as $t$ goes to $+infty$. We obtain that  $lim_{t o +infty}Y^2(Mimes N,[g+th])=2^{rac{2}{m+n}}Y(Mimes e^n, [g+g_e]).$   If $ngeq 2$, we show the existence of nodal solutions of the Yamabe equation on $(Mimes N,g+th)$ (provided  $t$  large enough). When  $s_g$ is constant, we prove that   $lim_{t o +infty}Y^2_N(Mimes N,g+th)=2^{rac{2}{m+n}}Y_{e^n}(Mimes e^n, g+g_e)$.   Also we study the second Yamabe invariant and the second  $N-$Yamabe invariant.