Second Yamabe Constant on Riemannian Products

2015/05/05 by Henry, Guillermo · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1505.00981

Abstract

Let (M^m,g) be a closed Riemannian manifold (m≥ 2) of positive scalar curvature and (Nⁿ,h) any closed manifold. We study the asymptotic behaviour of the second Yamabe constant and the second N-Yamabe constant of (M× N,g+th) as t goes to +∞. We obtain that lim_{t → +∞}Y²(M× N,[g+th])=2^(2)/(m+n)Y(M× \reⁿ, [g+g_e]). If n≥ 2, we show the existence of nodal solutions of the Yamabe equation on (M× N,g+th) (provided t large enough). When the scalar curvature of (M,g) is constant, we prove that lim_{t → +∞}Y²_N(M× N,g+th)=2^(2)/(m+n)Y_\reⁿ(M× \reⁿ, g+g_e). Also we study the second Yamabe invariant and the second N-Yamabe invariant.

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