Connected sum construction for $\sigma_k$-Yamabe metrics

TL;DR

This paper introduces a method to construct new Riemannian metrics with positive constant -curvature by connecting existing solutions via a connected sum, expanding the toolkit for solving the -Yamabe problem.

Contribution

It provides a novel connected sum construction for -Yamabe metrics, applicable to nondegenerate solutions when 2  < n.

Findings

Constructed new families of -Yamabe metrics

Extended the connected sum technique to nonlinear elliptic equations

Demonstrated existence of solutions for specific curvature conditions

Abstract

In this paper we produce families of Riemannian metrics with positive constant $\sigma_k$-curvature equal to $2^{-k} {n \choose k}$ 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 $2 \leq 2k < n$. The problem is equivalent to solve a second order fully nonlinear elliptic equation.