Connected sum construction for ${\sigma }_k$-Yamabe metrics

Abstract

In this paper we produce families of Riemannian metrics with positive constant ${\sigma }_k$-curvature equal to $2^{-k}\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ by performing the connected sum of two given compact {\em non degenerate} $n$--dimensional solutions $(M_1,g_1)$ and $(M_2,g_2)$ of the (positive) ${\sigma }_k$-Yamabe problem, provided $2\le 2k<n$. The problem is equivalent to solve a second order fully nonlinear elliptic equation.