Nodal solutions of the Yamabe equation and the second Yamabe constant

GUILLERMO HENRY

La Falda, Córdoba

Workshop; EGEO 2016, VI Workshop on Differential Geometry; 2016

FAMAF

Let $(M,g)$  be a closed  Riemannian manifold of $\dim(M)=n\geq 3$. We say that $u\in C^{\infty}(M)$ is a solution of the Yamabe equation if there exists $\lambda\in\re$ such that $u$ satisfies\begin{equation}\label{YE}a_n\Delta_g u+s_g u=c |u|^{\frac{4}{n-2}}u\end{equation}where $a_n=4(n-1)/(n-2)$ and $s_g$ is the scalar curvature of $(M,g)$.If $u$ is a positive solution of the Yamabe equation, then $u^{\frac{4}{n-2}}g$ is a Riemannia metric of constant scalar curvature $c$. It is well  know that there exists a positive solution of (\ref{YE}) if  $\lambda$ and the Yamabe constant of $(M,g)$  have the  same sign. A nodal solution of Yamabe equation is a solution of (\ref{YE})  which changes sign. The second Yamabe constant  is defined  by$$Y^2(M,g):=\inf_{h\in[g]}\lambda_2(L_h)vol(M,h)^{\frac{2}{n}}$$where $[g]$ is the conformal class of $g$ and $\lambda_2(L_h)$ is the second eigenvalue of the conformal Laplacian of $L_h$. This constant is related with nodal solutions of the Yamabe equation (see [1]).In this talk we are going to discuss some results about the asymptotic behaviour  of the second Yamabe constant of a Riemannian product $(W\times N,g+th)$ $(t>0)$.As a consequence, we are going to prove the existence  of nodal solutions for $(W\times N,g+th)$ (provided  $t$  large enough).Also, we will discuss the existence of $G-$invariant nodal solutions, where $G$ is a compact subgroup of the isometry group.The equivariant results are part of a work in progress joint with Farid Madani.