Second Yamabe Constant on Riemannian Products

TL;DR

This paper investigates the asymptotic behavior of the second Yamabe constant and related invariants on Riemannian product manifolds as a parameter tends to infinity, revealing limits and existence of solutions under certain conditions.

Contribution

It provides new asymptotic formulas for the second Yamabe constants on product manifolds and establishes the existence of nodal solutions for large parameters.

Findings

Limit of second Yamabe constant as t approaches infinity.

Existence of nodal solutions for large t when n ≥ 2.

Asymptotic behavior of second Yamabe invariants.

Abstract

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